Disclaimer: I’m not anyone associated with the show, I merely obsessively follow it.
The way I see it, there are three possibilities: River Song is either a character we have never seen before her debut on the series, a character we have seen fairly recently in “Who time”, or a character from the classic series. Let’s take those possibilities one at a time:
A character we have never seen before: Silence in the Library strongly suggested that she was The Doctor’s future wife, since she knew his real name and she aggressively flirted with him. I think that is a red herring, but it may be that is exactly the case: the obvious solution, applying Occam’s Razor, is the correct one. She is The Doctor’s future wife.
A character we have seen fairly recently in “Who time”: this is where things get interesting. A very interesting theory is that she is actually Amy Pond, in the future. She will have traveled with The Doctor for a while, potentially after Rory’s death (!) and she adopts a pseudonym… And becomes close enough to The Doctor to know his real name – or The Doctor knew he had to tell her his real name so his past self will trust her when he meets her for the first time – again. You know, “wibbly wobbly timey wimey” stuff… This is an interesting theory which I kinda like.
Another option: The Doctor’s Daughter, from the episode of the same name. This will let them potentially reintroduce the character and at the end of the episode she regenerates… Into River Song.
A character from the classic series: Lots of candidates here…
Romana: companion to the fourth Doctor and Time Lord/Lady. I like this idea, but I also like all ties that they establish to the old series, especially when they tie into my favorite Doctor.
Susan, the Doctor’s grand daughter. This is an interesting notion in that she probably knows The Doctor’s real name, and can add even more emotional resonance to River’s “death” earlier in the new series – we saw the death of The Doctor’s first companion.
The Rani: I don’t think River is a “bad guy” but this theory has made the rounds. I don’t like it.
So, that’s my overlong fanboy answer to this question. And, as I have so often been before, I’m probably completely wrong. But thanks for reading!
UPDATE: OK, we now know that, in the episode A Good Man Goes to War it was revealed that River Song is actually Melody Pond, Rory and Amy’s daughter, AND is also a Time Lord (because she was conceived inside the TARDIS) AND is also the little girl from the first two episodes of the season. I have another interesting theory as to where River’s story goes from here at this link.See question on Quora