The White Rabbit was a robot constructed by the planet computer of the Shore Leave Planet. The robot was a representation of a character from Lewis Carroll's "Alice's Adventures in Wonderland".
Although Dr. Leonard McCoy did not realize it until later, the rabbit was created in response to his thoughts about this story. The rabbit appeared before McCoy, commented that it was late, and then disappeared through a hedge. Moments later, a little girl appeared, asking a bemused McCoy whether he'd seen the rabbit. (TOS: "Shore Leave")
When the USS Enterprise revisited the Amusement Park Planet in 2269, McCoy briefly encountered the White Rabbit again. Like before, the rabbit was being chased by the same little girl. Later, McCoy and Sulu had a picnic lunch with Alice, the White Rabbit, and a dragon. (TAS: "Once Upon a Planet")
The White Rabbit was played by William Blackburn in "Shore Leave", who remained uncredited on-screen despite having dialogue; and voiced by James Doohan in "Once Upon a Planet".
According to dialogue from McCoy, the live action White Rabbit seen in "Shore Leave" and the animated one in TAS: "Once Upon a Planet" looked "exactly the same".