common burrowing rodent mammal, noted for prolific breeding, late 14c., rabet, "young of the coney," suspected to be from Walloon robète or a similar northern French dialect word, a diminutive of Flemish or Middle Dutch robbe "rabbit," which are of unknown origin. "A Germanic noun with a French suffix" [Liberman]. The adult was a coney (q.v.) until 18c.
Zoologically speaking, there are no native rabbits in the United States; they are all hares. But the early colonists, for some unknown reason, dropped the word hare out of their vocabulary, and it is rarely heard in American speech to this day. When it appears it is almost always applied to the so-called Belgian hare, which, curiously enough, is not a hare at all, but a true rabbit. [Mencken, "The American Language"]
Rabbit punch "chop on the back of the neck" (1915) is so called from resemblance to a gamekeeper's method of dispatching an injured rabbit. Pulling rabbits from a hat as a conjurer's trick recorded by 1843. Rabbit's foot "good luck charm" is attested by 1879 in U.S. Southern black culture. Earlier references are to its use as a tool to apply cosmetic powders.
[N]ear one of them was the dressing-room of the principal danseuse of the establishment, who was at the time of the rising of the curtain consulting a mirror in regard to the effect produced by the application of a rouge-laden rabbit's foot to her cheeks, and whose toilet we must remark, passim, was not entirely completed. [New York Musical Review and Gazette, Nov. 29, 1856]
Rabbit-hole is by 1705. Rabbit ears "dipole television antenna" is from 1950. Grose's 1785 "Dictionary of the Vulgar Tongue" has "RABBIT CATCHER. A midwife."