A rabbit. So called from the provincial word bun, a tail. The Scotch say of the hare, “she cocks her bun.” Bunny, a diminutive of bun, applied to a rabbit, means the animal with the “little tail.”
Bunny, lying in the grass, Saw the shiny column pass.
Bret Harte: Battle Bunny, stanza 1
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894