I really like the nickname [name_f]Gigi[/name_f] but, what are the full name possibilities? 
[name_f]Ginger[/name_f]
[name_f]Genevieve[/name_f]
[name_f]Georgia[/name_f]
Really a thing that’s starts with a G.
Technically, you can use any name that begins with “G” to get to [name_f]Gigi[/name_f] but here are some names which are usually associated with this nickname.
[name_f]Virginia[/name_f]
[name_f]Genevieve[/name_f]/[name_f]Geneva[/name_f]
[name_f]Regina[/name_f]
[name_f]Georgina[/name_f]/[name_f]Georgia[/name_f]/[name_f]Georgiana[/name_f]
[name_f]Giselle[/name_f]
[name_f]Ginger[/name_f]
[name_f]Giovanna[/name_f]/[name_f]Gianna[/name_f]/[name_f]Gia[/name_f]
[name_f]Ginevra[/name_f]
[name_f]Giada[/name_f]
[name_f]Gillian[/name_f]
[name_f]Giacinta[/name_f]
[name_f]Georgine[/name_f]
[name_f]Virginie[/name_f]
[name_f]Giorgia[/name_f] or [name_f]Giovanna[/name_f]
I have a pal named [name_f]Georgia[/name_f] that I call [name_f]Gigi[/name_f], which is a suggestion others have mentioned. I think it is the most natural name to extract [name_f]Gigi[/name_f] from (along with [name_f]Georgina[/name_f] and [name_f]Georgiana[/name_f]). The PPs were correct when they said basically any “G” name - or name containing “gi” anywhere - can be dubbed [name_f]Gigi[/name_f]. That said, I think the most natural way to go about it is for the full name to have a double “g” like [name_f]GeorGia[/name_f], therefore extracting GG and shifting to [name_f]Gigi[/name_f] 
[name_f]Gianna[/name_f] is another one that could work for [name_f]Gigi[/name_f].
I’ve often thought [name_f]Gigi[/name_f] would name a cute nickname for [name_f]Evangeline[/name_f].
[name_f]Giovanna[/name_f]
[name_f]Gianna[/name_f]
[name_f]Giselle[/name_f]
[name_f]Gioia[/name_f]
Giodana
[name_f]Ginny[/name_f]
Ginerva
[name_f]Gillian[/name_f]
[name_f]Giada[/name_f]
[name_f]Gina[/name_f]
Any name that starts with a G or has “gi” in it. [name_f]My[/name_f] favorites would be [name_f]Georgia[/name_f], [name_f]Genevieve[/name_f], or [name_f]Ginevra[/name_f]. Ooh, if you want to be daring, maybe [name_f]Eugenia[/name_f] or [name_f]Eugenie[/name_f]? Or are they still too clunky?