'Bunny' was originally a man's pet-name for a woman or a child. That is we called our ladies 'bunny' in much the same way as modern men now call them 'honey'. It has been a childish/pet name for a rabbit, too, since the 1600s.
I believe rabbit used to be the name just for the babies, and the adults were then called conies.
I think it changed because cony sounds rather too like a certain swear-word popular in former times.