The version of this song I am familiar with comes from Currence Hammons (WV) and he called it "Gettin there Rabbit."

The song I know as "Big Eyed Rabbit" comes from (I think) New England. Different words, different tune.

Yonder comes a rabbit, quick as he can run.
If I see another one, gonna shoot him with my double barreled gun.
gonna shoot him with my double barreled gun.

Big Eyed Rabbit's gone
(repeat 3 more times)