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 in Wonderland".
Although 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 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")