"large amphibious quadruped rodent of the genus Castor," Old English beofor, befer (earlier bebr), from Proto-Germanic *bebruz (source also of Old Saxon bibar, Old Norse bjorr, Middle Dutch and Dutch bever, Low German bever, Old High German bibar, German Biber), from PIE *bhebhrus, reduplication of root *bher- (2) "bright; brown" (source also of Lithuanian bebrus, Czech bobr, Welsh befer; see bear (n.) for the likely reason for this).
Formerly valued and hunted for its secretions (see castor) and for its fur, which was used in the manufacture of hats, so much so that beaver could mean "hat" from 1520s and continued so into 19c. even after they began to be made of silk or other material. They were hunted to extinction in Great Britain in the 16th century but have lately been reintroduced.
late 13c., "strenuous, ardent, fierce, angry," from Old French aigre "sour, acid; harsh, bitter, rough; eager greedy; lively, active, forceful," from Vulgar Latin *acrus (source also of Italian agro, Spanish agrio), from Latin acer "keen, sharp, pointed, piercing; acute, ardent, zealous" (from PIE root *ak- "be sharp, rise (out) to a point, pierce").
Meaning "full of keen desire" (early 14c.) seems to be peculiar to English. The English word kept a secondary meaning of "pungent, sharp-edged" till 19c. (as in Shakespeare's "The bitter clamour of two eager tongues," in "Richard II"). Related: Eagerly; eagerness. Eager beaver "glutton for work" [OED] is from 1943, U.S. armed forces slang.
"female genitals, especially with a display of pubic hair," by 1927, British slang, ultimately from beaver (n.1), perhaps transferred from earlier meaning "a bearded man" (1910), or directly from the appearance of split beaver pelts.
late 14c., "a beaver," from Old French castor (13c.), from Latin castor "beaver," from Greek kastor "beaver," literally "he who excels," also the name of one of the divine twins (with Pollux), worshipped by women in ancient Greece as a healer and preserver from disease.
It has been assumed that the hero's name was given to the animal because he was a noted healer and the odorous reddish-brown secretions of the inguinal sacs of the animal (Latin castoreum), were used medicinally in ancient times, especially for women's diseases. But the animal did not live in Greece in classical times (the closest beavers were north of the Black Sea), and the name probably was borrowed from another language, perhaps influenced by the hero's name. The Greek word replaced the native Latin word for "beaver" (fiber).
In English, castor is attested in the secretion sense from late 14c. Modern castor oil is first recorded 1746; it is made from seeds of the plant Ricinus communis but supposedly possesses the laxative qualities (and taste) of beaver juice.
"fur or pelt of the coypu," a kind of large, beaver-like rodent native to southern South America, 1836, from Spanish nutria "otter," also lutria, from Latin lutra "otter," an unexplained variant of PIE *udros (see otter).