Fame
 Warhol: 1Wh = 15 minutes  from Andy's statement that, "In the future everyone will be famous for 15 minutes."
Beauty
 milliHelen: 1mH = amount of beauty that will launch one ship
Happiness
 Bromeon: The amount of happiness Bromeo (sic) felt when climbing the balcony to meet his loved one, in the play by Million Shakesphere (sic).
 Hedon (or hed): The amount of happiness one feels when, after having walked 10 kilometers with a nail in one's shoe, the nail is removed.

 Both of these come from a story by StanislawLem, called Kobyszcz� or (probably) Hunt for Happiness. I read it in the Dutch translation of an English book called View from another Shore, which features stories from several European ScienceFiction writers. Something may have been lost in the triple translation, so if someone has access to a better source, please help me out here (AnswerMe).  AalbertTorsius
Magic
A Thaum is the basic unit of magical strength. It has been universally established as the amount of magic needed to create one small white pigeon or three normalsized billiard balls.

 from TerryPratchett. It is mentioned quite often in the Discworld Novels featuring Rincewind
Pratchett also invented a background character named Richter, who developed all sorts of ingenious measuring devices, such as the Rev Counter for use in ecclesiastical circles.
Speed
 FurlongsPerFortnight: 1 furlong/fortnight = 10 Snail's Pace (note: 1 furlong = 10 chains) This is incredibly (literally, for once) close to a centimetre / minute  0.99785914 cm/minute
 c (speed of light): 1.80 terafurlongs per fortnight
 c (speed of light): 18 terasnailpaces
Time
 kilosecond: 1ks = 16 minutes 40 seconds
 megasecond: 1Ms = 11d 13h 46m 40s
 gigasecond: 1Gs = 31.69 years
 terasecond: 1Ts = 31688 years (for comparison, the precession of the equinox is a 25920 year cycle)
 milliweek: 1mwk = 10.08 minutes
 microcentury: 1�C = 52.6 minutes
 nanocentury: ~pi seconds (to onetenth of one percent, 3.156926s)
 kilominute: 16 hours, 40 minutes
 megaminute: 1.9 years
 CuplaDays: amount of time to do any reasonable task
 Kalpa: One of the Brahmanic eons, a period of 4,320,000,000 years.
 Baktun: 144000 days, or 394 years in the Maya calendar; basically a Maya century if we consider they counted in base 20 and that 20^2 equals 400; 144000 in base 10 corresponds to {19}000 in base 20. The mass hysteria around 21 December 2012 was due to a millenialisticlike misinterpretation of the day the Baktun 13 began.
 6 or 8 Time Units of Random Duration: HatTrickSix? (opening the curious problem if one can determine properties for a random number bounded only by infinity...)
Force
 newton: weight of an SI apple (assumes you get 9.81 apples to the kilo (assumes that g=9.81 m/s where you are))
 nasa: either a newton or a poundforce  you choose
Distance
 picoparsec: 30.8 km or very approx 20 miles
 attoparsec: About an inch. `atto' is the standard SI prefix for multiplication by 10^(18). A parsec (parallaxsecond) is 3.26 lightyears; an attoparsec is thus 3.26 * 10^(18) light years, or about 3.1 cm (thus, 1 attoparsec/microfortnight equals about 1 inch/sec). http://www.catb.org/~esr/jargon/html/A/attoparsec.html
 football field: 360 feet (109.7 m) in American football, 100 meters in association football. Also doubles as an unit of area, so its exact usage is usually specified, as in "it's the length of 25 football fields" or "a surface the size of 50 football fields".
 Smoot: 5'7". http://www.catb.org/~esr/jargon/html/S/smoot.html
 Hubble: 1H = 10^9 to 10^10 light years  the approximate diameter of the observable universe.
 Hopper: distance an electrical signal will propagate in a copper wire in 1 nanosecond (see GraceHopper) This is approx 1 foot. [1 foot in vacuo, but signals move slower down insulated cable] "Lightfoot" is another word for nanosecond. See SpeedOfElectrons for more.''
 RCH: the smallest measurement known to man. The width of one "red c*nt hair".
Area
 Barn: 1b = 10^28 m^2 (as in, CantHitTheBroadSideOfaBarn?). the crosssectional area of an atom.
 Wales: 20,640 km^2, used in the British media as an equivalence unit of area, as in "An area of rainforest the size of Wales is cut down every year."
 Texas: 696,241 square kilometers (268,581 square miles). Texas is the largest contiguous state of the United States, and as a result it is used to measure surface in an appreciable order of magnitude. As in the Armageddon movie: "The meteorite is the size of Texas".
 Football field: 5353.36 square meters, or 57,600 square feet. Back in the nineties the media would often tout that rainforests were being destroyed at a speed of one football field per second.
 Shroud of Turin: approximately 4.84.9 square metres. See http://whatif.xkcd.com/70/ (note [2])
Volume
 Barn Hubble: 1bH = about a liter. An object one barn in crosssection would have to travel a thousand million lightyears before it traced out one liter of volume.
Frequency
 Terrahertz: The soft green light of Earth (green light (5200�) is about 577 THz; Green = Earth = Terra ~= Tera)
 "to bathe our wearied eyes in the green light of earth's bosom" ("The World of London, Second Series, Part I, Aristocracies of London Life", Blackwoods Edinburgh Magazine, Jan. 1843)
 "We pray for one last landing / On the globe that gave us birth; / Let us rest our eyes on fleecy skies / And the cool, green hills of Earth" (Rhysling, "the Blind Singer of the Spaceways", in "The Green Hills of Earth, Robert A. Heinlein, The Saturday Evening Post, 8 February 1947)
Bogosity
Complexity
Flow/Flux
 gallons per fortnight: Flow measurement used by Shaw in his 1984 paper describing the chaotic drippingfaucet experiment.
 Firkin per fortnight: almost exactly a sixpack a day.
Stench
 MilliBlatt  for definition and etymology see Hackers: Heroes of the Computer Revolution by Steven Levy (ISBN 0141000511 )
CPU speed: (well, not exactly)
Frequency:
 Barnum: 0.01667 suckers / sec
Poverty of Language:
Trauma:
Skill at playing basketball:
 The UseNet group rec.sports.basketball.pro used to measure the skill of NBA players in millijordans; one Jordan represented the amount of skill exhibited by Michael Jordan (in his prime). As no NBA player at the time (or even now) was as skilled as MJ, the millijordan was used instead. (No other information on how this scale was calibrated is available.) So would the symbol for that be mMJ? Or just J?
Rate of Speech:
 The Lampson. ButlerLampson speaks extremely quickly, and so this unit is named for him. As with many other units, one Lampson is quite large, and so most uses are in terms of milliLampsons. For example, ForrestBaskett? once gave a talk where he said "I'm starting to run short of time, so I will increase my speed to 700 milliLampsons."
 The Dirac, so named for PaulDirac. Equals one word per day, in reference to Dirac's tendency toward solitude and shyness.
Duration of Speech:
 milliStrom. Senator Strom Thurmond set a record in 1957 by filibustering the Civil Rights Act of 1957 for 24 hours and 18 minutes (although the bill ultimately passed); this duration (87,480 seconds) is one Strom. Therefore one milliStrom is 87.48 seconds, or very close to a minute and a half.
Quality of Literature:
 The Monkeyminute. It is famously said that an infinite (or a large) number of monkeys typing randomly at a keyboard for eternity, will produce the complete works of Shakespeare (or some other literary canon). It follows, naturally, that for works of lesser magnitude or quality, a smaller number of monkeys (or shorter duration) will suffice. DilBert once asked Dogbert to assess the quality of poetry he had written; to which Dogbert repeated the above assertion. Dilbert asked "what about my poem?" Dogbert replies: "Three monkeys, 10 minutes". Hence, the monkeyminute is a measure of literary quality.
http://www.dtc.umn.edu/~reedsj/monkeys.html
Academic Difficulty
 At my college we used a measurement unit named after the data transmission professor  to protect his identity, let's call it miliSerrano. One Serrano was the difficulty of his data transmissions course, one of the most notoriously difficult ones of the telecommunications engineering major. The encryption and security course was around 750 miliSerranos, i.e. not impossible but still rather tough. A filler course would be around the 100 miliSerranos, and a calculus course was around the 500 miliSerranos. Data transmissions, however, was not the most difficult course of the major  it turned out to be network management, which managed to reach the 1300 miliSerranos and still counting!
LAC
A distance "in terms of unexpected hassles creatively surmounted, wrenchingly difficult decisions made, and pits of despair climbed out of by the emotional fingernails ... on a par with any given day of the Lewis and Clark expedition."
NealStephenson:
CryptoNomicon
A humorous list seen circulating around the Internet via email:
 Ratio of an igloo's circumference to its diameter = Eskimo Pi ( => Ratio of an igloo's diameter to its height = Eskimo Two)
 2000 pounds of Chinese soup = Won ton
 1 millionth of a mouthwash = 1 microscope
 Time between slipping on a peel and smacking the pavement = 1 bananosecond
 Weight an evangelist carries with God = 1 billigram
 Time it takes to sail 220 yards at 1 nautical mile per hour = Knotfurlong
 365.25 days of drinking low calorie beer = 1 Lite year
 16.5 feet in the Twilight Zone = 1 Rod Serling
 Half a large intestine = 1 semicolon
 1,000,000 aches = 1 megahurtz
 Basic unit of laryngitis  1 hoarsepower
 Shortest distance between two jokes  a straight line
 453.6 graham crackers = 1 pound cake
 1 trillion microphones = 1 megaphone
 1 million bicycles = 1 megacycles
 365.25 days = 1 unicycle
 2000 mockingbirds = two kilomockingbirds
 10 cards = 1 decacard
 52 cards = 1 deckacard
 1 kilogram of falling figs = 1 fig Newton
 1000 grams of wet socks = 1 literhosen
 1 millionth of a fish = 1 microfiche
 1 trillion pins = 1 terrapin
 10 rations = 1 decaration
 100 rations = 1 Cration
 2 monograms = 1 diagram
 8 nickels = 2 paradigms
 2.4 statute miles of intravenous surgical tubing at Yale University Hospital = 1 I.V. League
"How often do you play for the school chess team?"  "With a frequency of about one microhertz."
See the first published paper by
DonKnuth,
The Potrzebie system of weights and measures, MAD Magazine 33 (June 1957), pp. 3637 (illustrated by Wallace Wood).
What is a potrzebie anyhow? Feed that word to Google and you get a lot of Polish websites. Unsurprising: it's a Polish word  means something along the lines of "necessary" (needed/required/wanted) modulo losses in translation.
 The word potrzebie used to be used in Mad Magazine as a nonsense word... it might, for example, be the answer given to any question that the answerer wished had not been asked. Knuth has cited Mad Magazine as his source for the word, and has also said that if he had realized at the time that people would some day want to know what DonKnuth's first publication was, he'd have published something else first.
The definitive answer: Mad publisher (in both senses) William Gaines encountered the word "potrzebie" on a the label of a bottle of Polish aspirin. Most Americans pronounce it potterzeebie, but the Polish pronunciation is nearer pohCHEByeh. If I recall correctly  from a conversation with a Defense Language Institute translator some thirtyeight years ago  it's the genitive form of a noun meaning need.
(Used in an instruction meaning "repeat dose as needed" or some such, I guess. Thanks for this detail!)
(Actually, not genitive, but dative or locative. From the probable context, I would guess locative: "in need", in the sense "when needed".
There are seven distinct cases in Polish.)
Also
http://www.strw.leidenuniv.nl/~vdmeulen/deeper/Articles/WeirdUnits.html
Knuth also (whimsically, as it happens) invented an yllion notation for big numbers (citation, anyone? "The Mathematical Gardner"). One hundred was 10^2, one myriad was a hundred hundred (10^4), one myllion was one myriad myriad (10^8), etc. So I guess that makes one centyllion equal to 10^(2^102).
Strictly, the ks, Ms and Gs aren't "whimsical", they are the official S.I. units derived from s. They really should be used in favour of those irrational minutes, hours, days, weeks and years. 
StephanHouben
They are used in
DeepnessInTheSky, by
VernorVinge.
That would be great if you lived in a basement and never went outside. Days are rather useful, corresponding to the patterns of light and dark in the
BigBlueRoom. Weeks are useful because seven things are easy to manage in working memory, and because if we had a lunar calendar they'd fit rather nicely into our months (with a little adjustment). 24 hours per day and 60 minutes per hour is a
wonderful idea, because being based on 12, days and hours are easily divisible by 2, 3, 4, and 6, rather than just 2 and 5 like those lame decimal systems (and 60 is divisible by 5 and 10 as well, but I don't need to tell you smart people that).
Actually, now that we have computers to do the arithmetic for us, a base10 system of measurements is kind of dumb. I would much rather have a base12 system that I can divide up by 3 easily. How about 60 cm in 1 m? 60 cents in one dollar? Then I could buy things that are 3 for a dollar without them ripping me off for that extra third of a penny!
Computers do the arithmetic, but you want a base that helps mental division? Shome mishtake, shirley?
It's not a mistake, and quit calling me Shirley.
It's a Vinge thing. His exwife, JoanVinge
?, used the same notation in her book,
The Outcasts of Heaven Belt (1982). 
EricJablow
My physics teacher used to say that when he grew up, volume was measured in
tumps and
shedfuls. I forget exactly how many tumps you need to get a shedful.
My chemistry teacher used to measure things in "chunks." A chunk was about half a mole, I think.
A while ago, a discussion of water pressure came up on a BBS I frequent, which ended with the increase in water pressure on a sheet of glass being measured in stone/square mesopotamian great cubit/fathom.
How do you measure fun?
MILIWIKI!!!
Well, a FunSize Snickers bar is, what, one ounce?
From
http://www.theonion.com/ 
Q: How big is your bong?
A: A thousand millibongs.
I once saw it asserted that there was an
official measure of insulation in use in the UK, of BTUs per square foot per centimetre per degree Kelvin. Or possibly Fahrenheit. It's a bit of a writeonly unit anyway.
It could have something to do with the BTU being defined as "one calorie pound degree Fahrenheit per gram Kelvin". "Pound per gram"; "Degree Fahrenheit per Kelvin"; you gotta love it. (I suppose it's better than just throwing in an arbitrary constant of 251.99576111111... which everyone would just approximate anyway!)
Dunno about that, but UK domestic gas bills give consumption in cubic feet, but charge per kilo Watt hour, via a conversion involving BTUs.
BTUs are used to calculate the required size of a radiator for a particular room volume, say. A DIY store's website may explain it. Another daft British unit is the kilowatthour. Let's follow this one through: it's rate of consumption of energy (kW) multiplied BACK UP by time to get back to energy...
No, it's pragmatic. Appliances are rated in kilowatts. This tells you how many units you'll use each hour you operate the appliance. In practice, that's pretty useful.
Why don't they use kilojoules, then?
cos then everybody would have to know that W=J/s and that is presumably too hard for some? Probably because of the factor of 3600 that creeps in when you convert from hours to seconds: running many home electrical appliances (televisions, room lights, ovens, etc.) for only a second or two might be construed as silly. Running them for an hour or two sounds more plausible: J. Random Householder would have an easier time of estimating how many hours an appliance typically runs for and wouldn't then have to convert that estimate to seconds to get a total power consumption.
A couple of informal units of measurement, which might refer to volume or mass or weight (I'm not sure) are
boatloads and
shiploads. A
boatload is enough material to fill up a boat, while a
shipload is enough to fill up an entire ship. Unfortunately I have often heard people use "buttload" instead of "boatload," and you may have already heard what they use for "shipload"...
Indeed I have  an odd variation on that is the "metric shipload"
I personally prefer the "metric fuckton"
Over the righthand side of the Altantic, we use
Shedloads. Quite a lot. Shedloads in fact.
EuropeanBillion vs
AmericanBillion
What's so whimsical about the Newton? It's the standard SI unit for force. And many other units are derived from it  the units for torque (Newtonmeter), work (the Joule is a meterNewton), pressure (a Pascal is a N/(m^2)), etc.
it's not the unit itself that is whimsical but its derivation from the weight of an apple that is whimsical, I have seen company press releases that actually did compare a force to a weight in apples where 1 apple happened to exert 1 N
WikiPedia has lots more at
http://en.wikipedia.org/wiki/Furlongs_per_fortnight.
CategoryWhimsy
CategoryTrivialBuffoonery
?