Even in Dutch they use the word konijn, I actually had no idea that "coney" was an English cognate for that word until reading this thread, I love learning these associations as making connections like this really helps in learning new languages.
I knew about Coney Island but I had never made the connection between Coney and Rabbit, I thought it was just a name.
Curiously there is another Isla Conejo / Rabbit Island near here too.
From Wikipedia:
It may be because when the Dutch people arrived in New Netherland, this island had many rabbits, so they named it 'Konijn Island', which became 'Coney Island', using the archaic term for rabbit in English.
As an aside, Spain itself was named after rabbits, which I find hilarious: http://www.ctspanish.com/rabbits.htm