Most people say it's Clair de Lune, but I think it's not. I've read Midnight Sun (Edward's perspective of Twilight) and there's a part that says that he's listening to Clair de Lune when he thought of the melody for the Lullaby. There's also a part that Alice helped him finish his composition and Esme loves it so much.

So which do you think is the real Bella's Lullaby? River Flows in You by Riyuma? or the one played by Edward on the piano on the movie? :)