also cony, "rabbit," c. 1200, abstracted from Anglo-French conis, Old French coniz, plurals of conil "long-eared rabbit" (Lepus cunicula) from Latin cuniculus (source of Spanish conejo, Portuguese coelho, Italian coniglio), the small, Spanish variant of the Italian hare (Latin lepus). The word perhaps is from Iberian Celtic (classical writers say it is Hispanic).
Middle English had two forms: cony, conny, also coning, cunin, conyng; Old French had conil alongside conin. Apparently the plural form conis (from conil, with the -l- elided) was taken into English and regularly single-ized as cony. The Old French form in -n was borrowed in Dutch (konijn) and German (Kaninchen, a diminutive), and is preserved in the surname Cunningham (from a place-name in Ayrshire). Rabbits not being native to northern Europe, there was no Germanic word for them.
Rabbit arose 14c. to mean the young of the species, but gradually pushed out the older word 19c., after British slang picked up coney as a punning synonym for cunny "cunt" (compare connyfogle "to deceive (a woman) in order to win sexual favors"). The word was in the King James Bible (Proverbs xxx.26, etc.), however, so it couldn't be entirely dropped, and the solution was to change the pronunciation of the original short vowel (rhyming with honey, money) to rhyme with bony, stony. In the Old Testament, the word translates Hebrew shaphan "rock-badger."
Association with "cheating" is from coney-catcher, "A term made famous by [Robert] Greene in 1591, and in great vogue for 60 years after" [OED]
CONY-CATCHER. A sharper, or cheat. Minshew has well expressed the origin of the term: A conie-catcher, a name given to deceivers, by a metaphor, taken from those that rob warrens, and conie-grounds, using all means, sleights, and cunning to deceive them, as pitching of haies before their holes, fetching them in by tumblers, &c. [Nares, "Glossary"]
Also 16c.-17c. a term of endearment for a woman. Coney-wool (1714) "fur of rabbits" formerly was much used in making hats, etc. Coney-hole "rabbit hole" is from mid-15c.