> Coney Island gets its name most likely from the Dutch settlers who called it Conijnen Eylandt or Konijn Eylandt, meaning "Rabbit Island", because the area once had a large population of wild rabbits
> the similarity to cunt (and particularly homophony with cunny) led through taboo avoidance both to the word's displacement in the main by rabbit and bunny
Look at the spelling: it would have rhymed with 'honey'. It's a pretty common phenomenon: we (Americans anyway) don't talk about 'cocks' and 'asses' anymore, we have roosters and donkeys.
The OED has a pronunciation history:
The usual current pronunciation with long ō ( /əʊ/ ) seems to have become established during the course of the 19th cent., and may in part be a spelling pronunciation reflecting the rarity of the word in general use in standard English at this date, when it may have been most familiar to many from use in the Bible (and especially in the Psalms) as the name of a foreign animal. However, this pronunciation is likely to have been reinforced by the desire to avoid association with 'cunt' and related words (compare 'cunny'), especially in a religious context.
Shakespeare played with this intentionally, and repeatedly. He uses conies and cony in multiple plays. One example, Henry VI, Part III, act 1 (beginning of play) he stages a battle and the capture of one noble in the civil war in the scene where famously later he calls the queen dowager "she-wolf of France"... widow of his hero Henry V of Agincourt fame. Reading the play today it seems just a dry history and bunch of pompous nobles prattling on an on with allusions to ancient Greece and Rome etc. But on stage with raunchy actors, the word cony (meaning: coney, rabbit) is surrounded in this scene by about ten different sex and genitalia words. And in the middle of the scene -- where somebody is about to die -- they stop the historical military action and have a sex-word raunchy poetry slam:
"Lord Clifford. I will not bandy with thee word for word,
But buckle with thee blows, twice two for one"
Here it may be helpful to imagine peak Tupac vs Biggie, throwing shade.
On the one hand, Shakespeare has to get the play text as written approved by humorless puritanical, literally Puritan censors and the actual Archbishop. So he can't say the c-word or make jokes about giant dongs. But, on the other hand, he does. By leaving it to the innate intelligence of the audience to grok the double entendres even in the middle of a civil war. (There are many other examples in other plays that are less subtle, but this particular one is clearly written for the actors to ham it up on stage).
In one scene we get:
"And full as oft came Edward to my side,
With purple falchion painted to the hilt"
"I am your butt, and I abide your shot"
(Note here Shakespear is actually mocking queen Elizabeth's minister IRL, Robert Cecil, who was hunchbacked, by making this hunchbacked character Richard of Gloucester ie the later super-ugly evil king Richard III... who is so ugly women cross the street and dogs bark at him... ahem, prefer the company of men and abide their shot as the literal butt of the butt joke. Simultaneously, Cecil's IRL first cousin Sir Francis Bacon also preferred the company of his male servants and is perhaps the actual crown prosecutor responsible for everything from slander or libel to state executions. Another writer, the perfectly-named John Stubbs, recently had his right hand amputated for slut shaming the queen herself ("a gaping gulf"). So this is not dull. It's high stakes!
[Footnote: Separately, plays and allusions to Edward II or Richard II are always about male favourites getting special access to the king. This becomes scandalous during the reign of James, who was notorious for err granting root access privileges; this led to fair questions later about who was the biological father of Queen Anne's son Charles, then duke of York-- for whom New York is named ever since. James himself was the "butt" of the same jokes, as it was not clear who his father was either, and Henry IV of France mocked him as "Solomon" ie the son of David, his mother's Italian lover, David Rizzio, whose root-access was suddenly terminated in 1566.]
"made an evening at the noontide prick"
"My ashes, as the phoenix, may bring forth
A bird that will revenge"
(this is played straight these days but in Elizabethan times and in her actual presence, this is a triple double entendre where ashes is enunciated as 'asses' ; Phoenix is one of the queen's personal metaphors for herself, sewn into her clothes, printed in her designs, worn as a jewel etc ; and bringing forth the revenge bird is yes vanilla the mythical phoenix coming back from the dead... but bird in this era allows the actors/company to play this scene with bird=cock or bird=woman, which still exists in slang today. Here noted in passing when the bird is a Quail, that's always a prostitute joke, which then you want to look for dunks nearby about somebody Burning with crotch-Pox which is STD jokes again about Robert Cecil among others who have sexually transmitted diseases which are serious in era of no antibiotics... in the public theater with randos in the audience this is played for laughs, but in a private court performance with the queen or later king James present this is mocking specific courtiers whom everybody at court already knows have one or more super gross STDs)
"I will not bandy with thee word for word,
But buckle with thee blows, twice two for one."
Bandy from the French bander, 'to stretch/erect/thrust' & buckle, colloquially to couple and mate. This works because it's plausibly deniable for the censors and conservative audiences, but the actors can play it for raunchy laughs when they want. Like the entire scene. Note also this begs the question if the "two for one" is a threesome joke, and invites contemplation who was known for the best parties back in the day when two-on-one happened twice, or more exotically, whatever a fivesome is called (twice two 'for' one).
In case the subtext isn't clear enough or you don't want to believe Elizabethan Catholic poets liked to mock humorless Puritan censors, in rapid fire the scene continues with:
#prick
#finger
#wound (contemporary slang for vag.)
#thrust
#"spurn" which Shakespear also uses in some weird ways in other plays eg "spurn me like (your) spaniel" so could be a other raunchy slang word we have lost since
#so strives the woodcock"
#"so struggles the cony"
And the glorious: "triumph thieves upon their conquer'd booty"
See also the note at rabbit: https://en.wiktionary.org/wiki/rabbit#English apparently rabbits themselves are all French. "hare is indeed inherited from Proto-Germanic."