The New Partridge Dictionary of Slang and Unconventional English (2008) says:
have a cow to become emotionally overwrought; to lose control US, 1966.
Speaking of Animals: A Dictionary of Animal Metaphors (1995) by Robert Allen Palmatier says:
HAVE A COW to have a cow. To have an anxiety attack. Source: COW. WNNCD: O.E. On the TV show "The Simpsons," Bart Simpson says "Don't have a cow, man!" meaning "Don't get all upset about it." Bart is likening an anxiety attack to giving birth to a cow - a frightening thought. Normally cows are the ones that give birth to cows - i.e., bull calves and heifer calves. Compare Have Kittens.
WNNCD is Webster's Ninth New Collegiate Dictionary (1983) and O.E. means Old English, but the O.E. must apply to the plain word cow rather than the phrase. (The OED dates cow to Old English.)
This Yahoo Voices article - Idioms Unpacked: "Don't Have a Cow" - also claims it means to (not) give birth to a cow, which would be distressing for a human to do. It lists a number of references at the end, but I've not followed them.
A quick search of Google Books shows this snippet dated 1962 from Field and Stream, Volume 67:
"Oh, don't have a cow," Chip said confidently. "They just haven't begun to fly yet."
"If they don't fly soon," Andy insisted, "they're going to need landing lights."
(Care must be taken with Google Books' snippets as they're often mislabelled, but following the story text we find an advert for a "NEW 1963 book of homes", so it's likely from 1962 or 1963.)
Searching Subzin.com, the first film I found to use the phrase was Sixteen Candles (1984):
00:39:00 I don't know, Jake.
00:39:02 I'm getting strange signals. Well, they're not comin' from me.
00:39:05 Everything's fine. Don't have a cow.
00:39:08 Okay.
00:39:10 Just remember one thing.
Edit: Good timing, as the OED have just released an update to the dictionary containing the phrase for the first time. The first quotation is from a 1959 newspaper:
1959 Denton (Texas) Record-Chron. 26 Mar. 3/2
He won't let me watch rock 'n roll shows... He'd
have a cow if he knew I watched 77 Sunset Strip.
In Old Norse, Ratatoskr means "drill-tooth" or "bore-tooth". It is the name of a mythical creature, a squirrel that runs up and down the tree of life called Yggdrasil, acting as a messenger between two arch enemies: the great eagle and the terrestrial dragon.
According to Albert Sturtevant, "[as] far as the element Rata- is concerned, Bugge's hypothesis has no valid foundation in view of the fact that the [Old Norse] word Rata (gen. form of Rati*) is used in Háv[amál] to signify the instrument which Odin employed for boring his way through the rocks in quest of the poet's mead [...]" and that "Rati* must then be considered a native [Old Norse] word meaning "The Borer, Gnawer"
Wikipedia
One can therefore assume that the word rat stems from a description of its behaviour.
Best Answer
Dictionary discussions of 'boning up'
J.E. Lighter, Random House Historical Dictionary of American Slang (1994) reports that "bone up" emerged from an earlier student term "bone," which was in use by 1859:
The 1859 example that Lighter cites is from a long poem titled "West Point Life." At the point where "boning" appears, the poem has devolved into a series of punning couplets. Thus:
Mitford Mathews, A Dictionary of Americanisms on Historic Principles (1951) has the following entry for bone as a verb in the relevant sense:
Mathews gives as its citation for this meaning the 1862 quotation cited by Lighter above, but identifies it as coming from "G. C. Strong, Cadet Life West Point 198." Hence, all three of Lighter's earliest citations are explicitly connected to the U.S. Military Academy in West Point, New York, which makes a strong initial case for the notion that "bone up" originated as military academy slang.
With regard to the noun Bohn, Mathews has this entry:
As for the word pony in Hall's definition of Bohn, Benjamin Hall, A Collection of College Words and Customs, first edition (1851) has this definition:
Though the second edition (1856) of Hall has the entry for Bohn cited above, the first edition (1851) does not; neither edition has an entry for bone in any sense.
Mathews is incorrect in supposing that Sylvia Clapin was the first investigator of slang to draw the connection between bone (in the sense of "to study") and Bohn (the slang noun). J.S. Farmer & W.E. Henley, Slang and Its Analogues, volume 1 (1890) has this relevant entry for bone as a verb:
Farmer & Henley's entry for Bohn reproduces Hall's definition word for word. John Bartlett, A Dictionary of Americanisms, fourth edition (1877) has this brief entry for "To bone":
No such entry for bone appears in the third edition (1861) of Bartlett; and it's interesting that the preposition Bartlett chooses here is into not up.
Robert Hendrikson, The Facts on File Encyclopedia of Word and Phrase Origins, fourth edition (2008) has this entry for the term:
'Bohned' and 'bohning'
Mathews (above) suggests that (as of 1951) no early instances of Bohn as a verb had been adduced. But Google Books searches find a number of matches for bohn as a verb and for bohning as a gerund, in a seemingly relevant sense, going back to at least 1874.
One is from "Alma Mater's Lament," in The University Palladium (1874):
From "Inter Nos" in Hamilton Literary Monthly (October 1885), discussing "lack of support to college athletics and college publications":
From Addenda to Memorabilia of the [Hillsdale College] Class of 1887 (1947) [combined snippets]:
From "Some Points for Young Engineers," in The [Rensselaer Polytechnic Institute] Polytechnic (May 19, 1888):
From Nine Members of the Saturday Night Club, The Seventh Son: A Story (1888):
From Hamilton Literary Monthly (April 1890):
From an item titled "In the Museum," in The Harvard Lampoon (May 12, 1892):
From "A Senior's Soliloquy," in The [Cornell University] Cornellian (1893):
From "The Marking System," in Hamilton Literary Monthly (April 1893):
From an ode to calculus, in The [Lehigh University] Epitome (1897) [combined snippets]:
From W.M.W., "A Notable Reform," in Michigensian, Volume 3 (1899):
A much earlier instance of Bohned—with a significantly different meaning—appears in an item titled "American Rifling" in Punch, or the London Charivari (1851):
The slang use of Bohned here is not entirely clear, but I think it may mean "translated into English-language editions for which they receive no royalties." The main joke here is the double sense of "bring down"—as in bagging the animal being hunted and as in "reducing or harming."
Early instances of 'bone up'
From "Things Chronicled," in The [University of Michigan] Chronicle (October 27, 1877):
From Henry Flipper, The Colored Cadet at West Point (1878):
From "Things Chronicled," in The [University of Michigan] Chronicle (June 23, 1883):
From "American Aristocracy," in Life magazine (December 27, 1883):
From "Nemesis," in The Michigan Argonaut (January 23, 1886):
From "Annapolis Notes" (April 21, 1886), in The United States Army and Navy Journal (April 24, 1886):
Conclusions
"Bone up" appears to have developed from the simpler verb form "bone," as J.E. Lighter, Random House Historical Dictionary of American Slang, reports. Several early instances of bone or boning in this sense are recorded from the U.S. Military Academy in West Point, New York, starting in 1859. The earliest match for "bone up" (or rather "boning up") is from 1877, from the University of Michigan.
The argument that "bone" as a verb with the meaning "study intensively" originated as "bohn," a verb alluding to the many volumes of the Bohn's Classics Library series, is less far-fetched than it might at first seem. Books in that series were extremely common in U.S. colleges and universities by 1850, and a number of references to bohned and bohning in the sense of intensive study appear between 1874 and the early 1900s.
Google Books search results show that students at some schools (notably the U.S. Military Academy and the University of Michigan) favored boned and boning, while students at others (notably Hamilton College and some Ivy League schools) favored bohned and bohning. Since the words were used in the same sense regardless of which spelling the writer adopted, it seems fair to view them as variants.
On the other hand, the recorded instance of boning in 1859—15 years before the first recorded instance of bohning, in 1874—raises the possibility that the pun on Bohn Classics was an afterthought attached to the existing bone of unknown origin. It is also noteworthy that an 1862 instance from the U.S. Military Academy uses bone in the context of studying mathematics—not a subject where Bohn translations would be likely to have much relevance.
At the very least, however, we can say that bohn was in fairly widespread use as a verb for several decades before bone (and bone up) won out in popular usage and rendered bohn obsolete. The connection between bone and bohn is much stronger than I had expected it to be.