I'd use
Lucy, in fact, I nearly did.
Luce is hilarious - a
Luce woman? Sounds like a 19th-century insult!
Lucina was the goddess of childbirth, so I suppose
Lucina could be a nonce name. I find it too long and frilly, compared to
Lucy. In fact, I don't enjoy
Lucia,
Lucinda,
Lucille etc either -
Lucy can stand alone.