Bunny
early 17th century (originally used as a term of endearment to a person, later as a pet name for a rabbit): from dialect bun ‘squirrel, rabbit’, also used as a term of endearment, of unknown origin. bunny (sense 2 of the noun) dates from the early 20th century.
wiktionary
From bun(“rabbit”) + -y, though its ultimate origin is unknown. Together with rabbit, bunny has largely displaced its rhyme cony.
From Middle English bony, boni(“swelling, tumor”), from Old French bugne, buigne(“swelling, lump”), from Old Frankish *bungjo(“swelling, bump”), from Proto-Germanic *bungô, *bunkô(“lump, clump, heap, crowd”). More at bunion, bunch.
From Middle English bune(“hollow stalk or stem, drinking straw”), from Old English bune(“cup, beaker, drinking vessel; reed, cane”), of unknown origin. Related to English bun, boon(“the stalk of flax or hemp less the fibre”), Scots bune, boon, been, see bun, boon. Compare also bunweed.
bunny (plural bunnies)
From bun(“small bread roll”) + -y.
etymonline
bunny (n.)
pet name for a rabbit, 1680s, diminutive of Scottish dialectal bun, pet name for a rabbit, previously (1580s) for a squirrel, and also a term of endearment for a young attractive woman or child (c. 1600). Ultimately it could be from Scottish bun "tail of a hare" (1530s), or from French bon, or from a Scandinavian source. The Playboy Club hostess sense is from 1960. The Bunny Hug (1912), along with the foxtrot and the Wilson glide, were among the popular/scandalous dances of the ragtime era.