One of the mysteries of the term rabbit is that nobody knows the origin of the word. It originally meant the young of the rabbit, the adult creature being known as a coney. In America, the term rabbit was applied, not to rabbits, but to various kinds of hare. We can trace the word rabbit back to sundry Germanic languages which have a word robbe, but beyond that we cannot go. We don't know the root of the word and its literal meaning.
As for English coney, this comes from Latin cuniculus which is possibly derived from a Celtic word meaning 'little dog'.