I love Cordelia too, it's my #2 FN. It's so spunky and distinctive and shaped perfectly. I also like the (imaginary) connection between it and Latin cor/heart. And Delia's a nice nickname. So is Della. I love it.
Lulu cannot work on its own, although it is definitely really cute. I know a little Lulu! She is Lucia Marie (her last name ends in -ia also. I kind of wish her parents had just gone with Lucia Maria and got it over with, especially as she's Spanish). I don't see why it would need to be on its own when it can be a nickname for almost any full name that begins with an L or has a prominent L sound.