Ok, a couple of possibilities just occurred to me. I don’t usually spoiler speculation, but if you’re sensitive to that sort of thing, you might want to skip the rest of this post, just in case I’m right.
Whoever the Time Lass is, something’s got to happen to her. I don’t think she’s River, since River’s first meeting with the Doctor is the Doctor’s last with her, and the Doctor already knows all about her. But even if she is, I don’t think she’s going to spontaneously regenerate into a 48-year-old archeologist. It’s been implied, I think, that it takes a very long time for a Time Lord to grow up, and I don’t think they can just leave her to run around the streets of New York as an orphan. They also can’t kill her off; Moffat’s not that dark!
I also don’t think Moffat is going to saddle the show with a permanent child companion. It wouldn’t make sense in-story, either. The Doctor’s not exactly the parental type, and the TARDIS is really no place to raise a child. They could drop Amy and Rory off at home (or somewhere else) to raise the child as their own (especially if she is their’s), but I don’t see that happening either. If the child is really a young Time Lord, or has the powers of one, she should be raised on Gallifrey.
Which means one of two things. Either the Doctor becomes determined to bring back Gallifrey from the end of the Time War, resurrecting all of Time Lord society, just so a little girl has a home (which would be very much like the Doctor to do) or…
Or he can’t bring Gallifrey back, but he can send something to them. To their past, giving her enough time to grow up in relative peace before the start of the Time War. If he could somehow transport her through the Time Lock (like Rassilon projected the white-point star diamond to Earth), could send her back to someone who he knew would care for her. Maybe with some sort of a note attached to a personal seal that would prove who it came from. A note that might read[spoiler]Dear Doctor,
This is your … well, you could call her your granddaughter, in a way. Please take care of her. She comes from Gallifrey’s future, a future the Time Lords would do anything to prevent. Keep her safe.
Her name is Susan.
Yours,
The Doctor[/spoiler]