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.
insect of the locust family, 1784, American English (perhaps first used by John Bartram), imitative of the stridulous sound the male makes when it vibrates its wings. In the eastern U.S., a familiar sound of a summer night; the sound itself was more accurately transcribed in 1751 as catedidist.
[T]heir noise is loud and incessant, one perpetually and regularly answering the other in notes exactly similar to the words Katy did, or Katy Katy did, repeated by one, and another immediately bawls out Katy didn't, or Katy Katy didn't. In this loud clamour they continue without ceasing until the fall of the leaf, when they totally disappear. [J.F.D. Smyth, "A Tour in the United States of America," 1784]