How did the word "beaver" come to be associated with vagina?
Solution 1:
Etymology Online offers that beaver in the gynecological sense is British slang dating from 1927, transferred from earlier meaning "a bearded man" (1910), or from the appearance of split beaver pelts.
Solution 2:
Jonathon Green's sources (as cited in Brian Hooper's answer) notwithstanding, the limerick that appears in Immortalia: An Anthology of American Ballads, Sailors' Songs, Cowboy Songs, College Songs, Parodies, Limericks, and Other Humorous Verses and Doggerel (1927) runs as follows:
There was a young lady named Eva
Who filled up a bath to receiv-a;
She took off her clothes
From her head to her toes,
When a voice at the keyhole yelled "Beaver!"
This limerick wouldn't provide much insight into the origin of beaver as a term for—as J.E. Lighter, Random House Historical Dictionary of American Slang (1994) puts it—"a woman's pubic hair," if it were not related to another phenomenon of the early twentieth century: the street game of calling out "Beaver!" at the sight of a man wearing a beard.
Lighter mentions this game in an earlier definition of beaver:
3. a beard; (hence) a bearded man; ... {A game called "Beaver," in which points are scored for sighting bearded men, arose about 1922 in Britain; see O[xford] E[nglish] D[ictionary] S[upplement].}
'Beaver!' in the news
A search of the Elephind newspaper database confirms that this game was popular in England in the early 1920s. From "Cambridge Students to Wear Beards Forever: Beavers in London Pride Themselves Upon the Length of Their Whiskers," in the Columbia [Missouri] Evening Missourian (June 26, 1922):
London, June 26.—The Hirsute Half-hundred, those whiskered gentry who astonished London a few weeks ago with their slogan, "A beard on the chin keeps the shaving money in," have now been eclipsed by the Cambridge University student society, which has sworn to wear beards forevermore and are known as the Beavers.
They fall on all unbearded undergraduates on sight yelling, "Beaver! Beaver!" The unwhiskered have entered joyfully into the game and try to spot a Beaver before their fellows.
One Beaver, who boasted a twelve-inch beard, had it pulled off in one of these "rags." To the disgust of his fellow Beavers, they found it was a spoof beard.
From "English Lord Tells of Game of 'Beaver' Played with Boards," in the Cairo [Illinois] Bulletin (October 6, 1922):
New York, Oct. 5. (By A[ssociated] P[ress])—Lord and Lady Mountbatten—she is one of England's prettiest and richest women and he is King George's cousin—decided today they would go to the world's series [baseball] game and compare it with London's new outdoor sport—"beaver."
...
"Beaver" said Lord Mountbatten, "is a street game anyone can play. You walk along with a friend. If you spot a chap with a beard you call out 'Beaver'. That counts 15 points. If it is a white beard, this is 'polar beaver' and counts 30. You score as in tennis. The winner makes the loser buy the drinks. And it is driving beards right out of London."
After reprinting the AP story on October 7, 1922, the Roanoke [Virginia] World-News followed up exactly a week later with a different account of the scoring rules for "Beaver":
It was Lord Mountbatten, who told of London's new outdoor sport, 'beaver,' and now a young Englishman has written a friend in this country giving the rules of the game. Here they are:
Each player on sighting a beaver should call out "Beaver" and score one point.
A player may huff another for not seeing a beaver and count four points.
A beaver who seeks concealment in his overcoat may be called, the player calling him to count three points.
Beavering of foreign visitors does not count. This is a rule, but it is never carried out.
In cases where there is uncertainty as to whether a person is beavering or merely unshaven he must be passed over, but marked down for future chukka.
Calling "Beaver" during a golf stroke is forbidden.
A person who calls "Beaver" and finds that he has made a mistake forfeits two points.
Actors may be beavered unless it is stated on the program that the beard is false.
From "Why King George of England May Have to Lose His Beard: How the Game of 'Beaver' Which All England Is Playing Is So Threatening the Proper Reverence for the Throne That Banishment of the Royal Whiskers Seems Imperative," in the Washington [D.C.] Times (October 22, 1922):
Now, in some unknown way, it has happened that by common consent the sight of any member of the royal family who wears a beard counts a full set and gives the player who first spies the royal whiskers the game. One observing these whiskers he or she calls out "Royal Beaver!"
...
Of course, in the case of the King, people playing the game do not usually rudely shout "Royal Beaver" when their eyes fall upon him. But they do convey the discovery by signal. On at least one occasion, however, the shout jarred in most unseemly fashion upon the royal ears. The King was visiting recently at Cambridge, and as he entered the hall where the students were assembled almost the whole undergraduate body arose to their feet with the shout of "Royal Beaver! Game, set, match!"
...
One thing must be noted. The game still has an element of politeness or at least chivalry. A "bearded lady" is not a "Beaver," and if any one so far forgets himself as to call "Beaver" in such a case he is barred from playing for a month.
And from "Beaver! Beaver! Score Twenty: Nice Little Game Can be Played by Two People—Teaches One to Be Observant Always," in the Monmouth [Illinois] Daily Atlas (October 28, 1922), which asserts that the game was invented by a humor columnist from the Chicago [Illinois] Tribune:
Locally the game is played mostly by college students and The Atlas warns its male readers not to be frightened or take offense when some one points at him and yells "Beaver."
It is argued by the proponents of the pastime that it increases the player's powers of observation and teaches him to keep his eyes open. Whether this is true or not is not known but no one will deny that it is a very entertaining amusement and local fans are waiting to see if the game which is gaining a foothold nationally, will become popular in Monmouth.
A followup article in the Washington Times on December 3, 1922, titled "Beavers, Which Flourished Here in Mid-Victorian Times, Now Face Extinction; Rarely Seen on Streets of Capital," reports that beards had become quite scarce in Washington, D.C., as well as in England in recent days.
A pro-beard backlash against the game of "Beaver" is noticeable in the pages of the [Adelaide, South Australia] Register (December 22, 1922) and a counter-backlash in the pages of the [Gawler, South Australia] Bunyip (December 29, 1922). After a couple of additional incidents reported in South Australia—in the [Adelaide] Chronicle of March 3, 1923, and in the [Adelaide] Register (again) of March 27, 1923, the Elephind matches for "out beaver" very nearly stop. A last article, "Beards Out of Favor: Even Farmers Discard Them," in the [Adelaide, South Australia] News (September 1924) lays out the demise of the beard:
Beards are not what they were. A leading hairdresser in Adelaide said so today. In company with everything else in the world they are slipping back. Even the country folk, formerly their chief supporters and advocates, have rejected them.
...
A look round the leading saloons revealed smiling, beardless countrymen waiting their turns to be shaved.
It looks as though the small boy of the future is to be robbed of the privilege of cheekily yelling "Beaver."
But memory of the fad was very much alive in the fateful year of 1927. From "Parliament: From the Gallery: A Sitting Short but Not Sweet" in the Sydney [New South Wales] Morning Herald (January 12, 1927):
Depressed by the lassitude of a pitifully short vacation and the damp heat of this scantily ventilated room, members just spread themselves out and breathed heavily. You might have heard that muted, pianissimo hiss before you entered the gallery, and you might have thought, listening for a moment, that everyone was conspiring together for some frightful purpose: planning to call out "Beaver!" to Mr Lysaght perhaps, or to embarrass Mr McKell with the rare and unexpected difficulty of a question. But they weren't conspiring, or thinking, or caring at all about anything, apparently, ...
And from Mercurius, "Passing Notes," in the [Hobart Tasmania] Mercury (October 22, 1927):
"Mercurius" inclines to think that these "scalers" [of out-of-date tram tickets] are the stuff that dreams are made of, like those angelic beings who "scaled" Jacob's ladder. But to test the question, can we not invent a game like that of "beaver," so much in vogue among English gamins a few years ago? Instead of calling out "Beaver" whenever we spy a hapless wight with whiskers, let us shout "scaler" whenever detect one of this host of miscreants. What fun it would be to ride our trams then!
This is the same year that the limerick about "a young lady named Eva" began to appear in limerick collections. It seems clear that the joke in the limerick is that someone is playing the old "Beaver" game but applying it to the wrong hair on the wrong part of the wrong anatomy.
Conclusion
It is quite astonishing that the "young lady named Eva" limerick's takeoff on the street game of Beaver—a game that seems to have hastened the demise of the beard during the early decades of the twentieth century—has evidently had far more cultural staying power (as judged by popular slang) than the original game itself.
It's also noteworthy that the first criers of "Beaver! Beaver!" were bearded (and stridently pro-beard) college students at Cambridge in 1922, who used the refrain as a battle cry as they harassed unbearded undergraduates. The tables certainly turned soon enough on that front.