The word bunny has been with us for some time. Etymonline dates it to the late seventeenth century Scottish dialectal 'bun'. It was suggested to me, by – I think – Joe Cremona (whom I have mentioned several times before [and that's what meta data is for!]) that prim English nurses and governesses were keen to discourage their charges from using the word descended from the Latin cuniculus – whence come the Spanish conejo, Portuguese coelho, Catalan conill, and various other words including the English 'coney' (rabbit fur).
The word coney was used as a rhyming euphemism for the 'female intercrural foramen', as – like honey and money – it did rhyme with cunny (Latin cunnus in which the double n yields, as often the Spanish ñ – whence  coño). Perhaps the stressed vowel in our 'coney' changed from /ʌ/ to /əʊ/ for euphemistic reasons similar to those nursemaids'. I have not met the word cunny 'in the wild', but it was used in the script of The Unforgiven in that anatomical sense.
But I must get on. I want to get OA done by the end of the week.
Added to last (full) para.
PS I assumed that, among those words derived from cuniculus, I might include Italian coniglio. But I wasn't sure at the time. I have looked it up, and it turns out that I should have had the courage of my convictions. But the dictionary yielded two other bits of information:
- coniglio can also be used in Italian as a taunt to someone who's not brave: 'chicken'
- 'bunny' is coniglietto
Le bon Saint EloiWhen I heard this [RIP Cedric Baring-Gould, a brilliant French teacher, ahead of his time, under-estimated by almost everyone], the irony was lost on me.
Lui dit <o mon roi
Est bien essoufleé. >
<C'est vrai – lui dit le roi –
Un lapin courait après moi. >
Coniglietto is striking for a more formal reason. It has the Italian diminutive suffix -etto/a (as in many borrowed words familar to English speakers: libretto, cornetto, bruschetta...), but coniglio is itself derived from a Latin (-ulus) diminutive.
Great have little fleas upon their backs to bite 'em,
And little fleas have lesser fleas, and so ad infinitum.