I like Renesmee's name, don't get me wrong :)

But if I truly wanted them to name her something else, I always think of 'Elizabeth'. Edward's biological mother was named Elizabeth. She looks like Edward, and evidently strongly loved her son. According what Carlisle has told Bella, Elizabeth would try to nurse Edward when he was sick with the Spanish Influenza, even though she was ill herself. Then, right before Elizabeth passed away, she told Carlisle to save her son by changing him into a vampire. Carlisle thought about this, and of course fulfilled Elizabeth's wishes :)

Maybe Edward and Bella could've named Renesmee after Edward's mother, because Elizabeth seemed like an amazing woman. If it wasn't for her, so many things wouldn't have happened; Carlisle wouldn't have changed Edward, Carlisle wouldn't realize that he has the ability to change people in order to save their lives instead of killing them, then he wouldn't have changed Esme, he wouldn't have changed Rosalie or Emmett. Alice and Jasper would have never found them because what family would there be to find? Lastly, Bella would have never met Edward and fell in love with him. She wouldn't find the family she always belonged with, and Renesmee's birth (and Jacob's soul mate, although I disagree with that), would have never happened. I don't even think Bella would have been born if it wasn't for Elizabeth, because Bella and Edward were meant to be together :)
And so many other things that I can think of....

You see, a lot of this happened because of Elizabeth. If it weren't for her, the Cullens wouldn't have existed.

I think naming their daughter after Elizabeth would be a sign of honoring this woman, and acknowledging what she has done for all of them. :)

It's their choice, if they like it then so be it. I never really understood what's the wrong about this name, it isn't that bad. So what if it's not a real name, vampires aren't supposed to exist either...

I love the name Renesmee!

Marie or Elizabeth had be alot better. Renesmee sound really ugly.

Renesmee is unique, one of a kind, so it is her name.