There's the ''genie in a bottle'' association, but any language student would know that this is what a girl that was found at the age of 13 who hadn't learned to speak or write because her parents kept her isolated from speech and beat her up when she made a sound was publicly called. So, the name doesn't have the nicest asociations to me, and I think the spelling Jeanie would be much prettier. However, this should only be a nickname anyway, not a proper name.
I can't help but think of the song Genie in a Bottle, and the name itself might lead to teasing. It's hard to believe that this silly nickname is short for the beautiful name Eugenia.