All rabbits are baby rabbits.
English and French both lost their early words for ‘rabbit’ because they sounded rude, and replaced them with words for ‘baby rabbit’.
The source of the problem, in both cases, was the Old French word coniz, which became French connin and also, thanks to the Norman invasion, English coney.1
Now coney (or cony) is usually pronounced to rhyme with ‘bony’ these days, but that's actually a modern innovation. Historically it rhymed with ‘bunny’, just like similarly-spelled words such as honey or money.
Of course, this meant it sounded exactly the same as cunny.
This could be a problem when some of those Bible passages2 about how ‘the coney ... is unclean unto you’ were read out loud.
The fact that the word was becoming less common was therefore a bit of a godsend. Though even in 1838 (according to the OED), Smart was writing that ‘it is familiarly pronounced cunny’, but that cōny ‘is proper for solemn reading’.
The word rabbit, which already existed3 and meant ‘bunny, baby rabbit’, was therefore pressed into service. Which I suppose is one of the reasons we no longer have a good word for a baby rabbit.
Compare that with the French situation (if you haven't fallen asleep or shot yourself yet). Their word for rabbit, connin, unfortunately bore an uncanny similarity to the word con, whose meaning you will already have inferred. If you read some of the more insalubrious Old French fabliaux (which is basically what I've been doing all week), you'll be amazed at how often a rabbit crops up in the story, just so they can get a good muff gag in.
Con is not, as far as anyone knows, related to cunny, by the way, so it's kind of an interesting coincidence that the same problem cropped up in both languages.
The French solution was similar. They had this word floating around for a baby rabbit - lapereau. The -eau bit means ‘baby’, so they took that off, and stuck on the ending from connin, giving them a new word which seemed to mean, by analogy, ‘grown-up rabbit’ - lapin. So on both sides of the Channel (or Quebec border) we're using words for ‘baby rabbit’ because the grown-up ones were too embarrassing. This is a nice thought.
And if you're wondering what we called rabbits before 1066, the answer is, nothing4. As well as bringing the word coney with them, the Normans also brought the first actual coneys with them, to breed for meat. Welsh cwning, Irish coinnín, Gaelic coinean are all borrowed from Middle English. There's no mainland Germanic word for them either, and the source of cony, Latin cuniculus, was apparently borrowed from some Iberian language - which is one of the reasons we know that rabbits are native to south-west Europe but didn't make it further north till considerably later.
Some tangential notes
1. Actually a back-formation from the OF plural, but that's another story.
2. Although technically, when the Authorized Version uses ‘coney’ it's translating a Hebrew word which probably referred to the rock hyrax. You can't eat them, Leviticus says so.
3. I don't know where that comes from though. Stop reading the notes and get back to the text
4. Although. I wonder how the relevant OE Bible translations go? I'm not sure if there is an OE version of Leviticus, but maybe weofodthignen can enlighten me (when she gets around to reading this in about 6 months)
5. There is no 5. Look again