As an aside, Spain itself was named after rabbits, which I find hilarious: http://www.ctspanish.com/rabbits.htm
This is of limited value in learning English; at least in America, I wouldn't expect the man on the street to know the word at all.
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.
Tangent, but in line with what others have commented, American English tends to pronounce the first syllable with a long o (like saying the letter 'o'), whereas British tends to pronounce it with a long oo sound (like the vowel sound in the the word 'food'). I've never heard it pronounced with a short 'u' sound (an 'uh' sound), but others intimated they either have, or have heard it pronounced close to it, which is interesting.
The Spanish word is pronounced more like "koh-neh-hoh" (or [ko'neho] in IPA), no "long a" ([ei] in IPA) there. The English "long vowels" (diphthongs really: a combination of two vowel sounds) would be written with two letters in Spanish (and IPA).
Three out of five are diphthongs. "Long E" is /i/. "Long u" is /u/.
https://en.wikipedia.org/wiki/Diphthong#English https://en.wikipedia.org/wiki/English_phonology#Vowels
Edit: I may be assuming an incorrect pronunciation of coney. Even as a native speaker English sucks.