Carter Burwell who composed "Bella's Lullaby" shares the experience with us on his website.
He has this to say about the Lullaby:
"And so this unnamed tune became "Bella’s Lullaby." It's worth noting, though, that neither Catherine nor I ever called it that. We always referred to this tune as the "Love Theme" and I think it makes much more sense if you think of it this way. It's more complex and emotional than any lullaby I've ever heard."
Here's what he says about Rob:
"This case was unusual in that Rob is a fine musician, and fully capable of playing the piano or probably any other instrument."
Go here to read this fascinating & in depth article on how he came up with "Bella's Lullaby."
Source: Thanks to Elaine Turner