My husband's newest obsession seems to be Kit for a girl, and yes as a given name.
I love Kit!
Actually, it might be the only name that I feel works equally great on both genders.
I think Kit is great!
I'll admit that, as a given name, Kit seems more masculine to me.
I think the given name "Kit" lends itself to easily becoming "Kittie" which is rather cat-tish!
I love the name Kathleen and Kit works well as a NN for that.
Kat is also cute as a NN. You could use Katrine, Kathleen, Katya . . .