There is also Genny and Gennie. Sounds like Jenny.

Trivia for you: a female donkey is a 'Jenny' Don't worry though, no one will think that you're a donkey ;) Use the spelling that you prefer. It's a very pleasant sounding name.

I don't think there's any rule or any strong predominance. "Jenny" and "Jennie" are (usually) nicknames for "Jennifer," although it can be given as a name in itself. In the U.S., women often exercise creativity and decide for themselves how to spell their own nicknames. For example, Barbra Streisand uses a personal variation of "Barbara." "Eleanor" and "Elinor" are both common. I personally know a Jennie and a Jenny, as well as a Jena and a Jenna. "Jenni" is common, too. If you are thinking of this as a name you want to use yourself, I strongly recommend that you just choose the one you like best. People often choose the spelling to match that of a famous person they admire, by the way.