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I think Didi and Gigi are fine. Gigi for pretty much any name that starts with Gi-, like Ginger, Ginevra, Georgia, Gillian etc, and Virginia. I'm not a big fan of initial-nicknames. G.G. is just that, it's not Gigi.
Mimi is so-so. Not my style really, but it's alright.
Fifi is ... well, probably not something one should call a child from the start. It's something a person would grow into by accident because it suited her.