I own nothing.
There are memories of Melody Pond, of course—however brief they are, memories are forever and they are stamped on the mind. Melody Pond may exist only in memory but she does exist.
In memories there are three people who love her more than anyone else in the universe. Her parents for two, and the Doctor for another. They in their minds all think about how she will be raised, how they will always protect her and never let anything happen to her.
Melody Pond will be a superhero, if Amy has anything to say about it. When she gets her baby back to Earth Melody will have the kind of childhood Amy couldn't (at least in a universe that was starting to crack and shatter): a childhood with both her parents alive and well, ready to guide and nurture her.
Rory would be happy if his daughter was a geography teacher, so long as she lives long enough to see that day. Rory will move heaven and earth to protect his daughter and God help anyone who gets in his way. Though he just wants to go back to being a nurse Rory will gladly become the Last Centurion once again for the sake of his daughter.
The Doctor… The Doctor doesn't know quite what to think of all this. He supposes that, in some informal sense of the word he's Melody's godfather. If Amy and Rory let him there's a good chance he'll show up from time to time to whisk her off somewhere exciting and new over the weekend.
But this doesn't come to pass.
You see, Melody Pond died. Her flesh didn't perish but she did. Melody Pond was dead from the moment the baby Amy held in her arms dissolved and spilled to the floor.
There's only River Song now. River Song is a child of war; River Song is a child of cold steel and hard places. River Song is never a child in the sense of what a child can be. She can hold a gun before she can write; she can kill a man with deadly ease before she's old enough to have a child. River Song has never known the touch of a mother or the warm protectiveness of a father or the good-natured indulgence of a godfather. She is a weapon before she is anything else, and River Song can never be what Melody Pond was.
Only in memories can she ever be the child Rory, Amy and the Doctor loved.