Kit registers 62 times at rank 1649 on female pets, though the registry's female skew may partly reflect the name's historical use as a diminutive of Katherine. Kit is one syllable, gender-neutral in practice, and has a nimble, quick quality that suits animals who move that way.
The Name's Many Sources
Kit arrives from multiple directions: a diminutive of Christopher or Katherine, the word for a young fox or rabbit, a term for a set of tools. As a pet name, the baby-animal definition adds a layer of appropriateness. Calling a young cat or rabbit Kit is quietly etymologically correct in a way that most owners don't intend but would appreciate. The human name is at /names/kit.
The Pop-Culture Thread
Kit Harington from Game of Thrones brought the name into wide contemporary visibility, though the pet-naming demographic is probably drawing as much from Kit Kittredge (the American Girl character) and the general nimble-animal association. The name has cultural currency without being obviously borrowed from any single source.
Sound and Breed Fit
KIT is one syllable, clean and precise. It calls without effort and suits small, quick-moving animals especially well: cats, small terriers, Whippets, and ferrets or rabbits where the etymological accuracy is a genuine bonus. For dogs, Basenjis carry the compact energy well.