It's admittedly quite an interesting name, and it doesn't sound far too pompous, megalomaniacal, or foreign, but it might lead to bad nicknames like Iffy and Genie. It also sounds a bit elitist, but I suppose some might say the same about the gorgeous Phaedra, even though it's much simpler. I wouldn't talk you out of it.
This is such a gorgeous name - for a fictional person, anyway. I really can't see an Iphigenia walking around without a good, solid nickname - maybe call her by her middle name, since nicknaming options are scarce here.