I think you should name her Claire, this is coming from a Claire Elizabeth, LOL. Figures...I think a REALLY pretty combo is Marie Claire, or Claire Marie, it could be hyphenated. I LOVE my name, and I never ever ever get sick of it. My mom chose it because it's simple, nickname free, and has a beautiful meaning. Here are some options:

Claire Elizabeth

Claire Anne

Claire Louise

Claire Marie

I hope you deeply consider this beautiful name.

here are some suggestions for names to go with Claire:

Claire Louise
Meredith Claire
Emily Claire
Sophie Claire
Madison Claire
Claire Isabelle
Mary Claire
Rachel Claire
Summer Claire
Emma Claire
Claire Pruedence
Lyra Claire
Catherine Claire
Joan Claire
Abigail Claire
Olive Claire
Violet Claire

hope that this help, tell if you like someone

I have # friends with double names

Samantha-Kate
Kara-Mae
Trisha-Mae

(suggestions)

Mary-Claire
Malia-Claire
Malyssa-Claire

(m and c's click together, don't they?)

Celynne-Claire
Selena-Claire

or if not double first names, how about double middle names?

My friend Gael kids have double middle names and they go like this:

Meredith Erin-Lyra
Amara Miki-Evelyn

Aiden Noah

Mercy Mae-Ann

hope this helps!

What about a play on Claire?
Clairice
Claira
Clairene
Claira-lee
Clairy

There is a really pretty Irish actress named Mariclaire Costello. These could be hyphenated or spaced.
Also-
Ava Claire
Amy Claire
Anna Claire
Bethany Claire
Brenna Claire
Dana Claire
Eva Claire
Gina Claire
Hannah Claire
Ivey Claire
Jessi Claire
Jenna Claire
Jonna Claire
Megan Claire
Nena Claire
Rosa Claire
Sara Claire
Tessa Claire