Alright, this is a decision I'm leaving entirely up to you, so cast your votes on my profile on FF.net.
The REASON: When I had originally created LOAS, I used the name Renee (which is my middle name). I've always been horrible when it comes to creating names - but the more I look at it the more unprofessional it seems (although most of you probably wouldn't know my middle name). People have asked me before if Renee is a self-insert, which she isn't. I'm not even half that cool.
I finally decided upon a name that works: Natalie Klein. It still allows for a nickname to work "Nattie" in place of "RenRen" for Renee.
Her name would be the only thing that would change. I would simply go back over the chapters for both stories and change her name. There would be no changes whatsoever to her character's personality. It's the same character, just with a different name.
So vote. As I said, it's entirely up to you.