WARNING: If you are not a Doctor Who fan, you will not understand any of this.
Also, if you are a Doctor Who fan with no idea of who either River Song or Romana is, you will not understand any of this.
Disclaimer: this post will be edited continuously as I gather more info and musings. I am a comparatively new Whovian and I don't know a whole lot, so forgive my naïveté.
I've seen many ridiculous theories on the identity of River Song, such as "She's Jenny!" or "She's Amy Pond!" or whatever else nonsense people can think of.
My guess: Romana.
Why isn't it Jenny?
Jenny is the Doctor's daughter, generated from a piece of his DNA (against his will). She has no idea who or what she is, she just knows the Doctor is her "father" (by a lucky coincidence, Georgia Moffett is the daughter of the Peter Davison, the 5th Doctor), a lot about combat and gymnastics, how to walk and talk, and how to seduce (isn't that how they got out of the jail cell?)
Why would a daughter call her own father "sweetie" or "my love" and speak as if they were equals? It just makes no sense. Don't even dare bring up incest as an argument.
I think most people already get the hint that River is the Doctor's wife from the future. This has basically been confirmed in the scripts and by Moffat himself. It is quite obvious that their relationship is markedly different from a father-daughter relationship. I have no idea why people keep insisting it's Jenny. Sheesh people, think!
Why isn't it Amy Pond?
Because Amy Pond's already married and she's a completely different person. She and River Song have basically nothing in common, even superficially. I wouldn't call her hair "red" or her accent "Inverness Scottish." This is a bit weaker as a counter-argument, but I just don't think it's her. The connection of "Pond" and "River" as bodies of water has already been dismissed by Moffat.
Why isn't it the Master?
Regenerations aren't known to be gender-switching processes. It may be a black swan, though. In any case, I don't think such an intense rivalry could turn into romance.
Why do people think it's not Romana?
They cite various reasons like
1) Time Lords can sense each others' presence
2) Romana and River are two very different people
3) Romana was locked away with all the other Gallifreyans in the Great Time War
4) River doesn't regenerate in Silence in the Library/Forest of the Dead.
1) There have been many instances when the Doctor didn't recognize other Time Lords (I can't think of one right now, but I remember making note of that fact). Also, what about the Chameleon Arch? What if she wanted to disguise herself to prevent "spoilers," which is something she presumably knows that the Doctor hates?
2) Who's to say that Romana didn't regenerate and become a new person? Of course personalities can change with regenerations, in fact, it is known that she became a radically different person when she was Lord President of Gallifrey.
3) No one ever knows what happened to her before the end of the Time War. It is known that she was replaced by Rassilon as Lord President. Maybe she regenerated again and this time decided that the Time War was a lost cause and went off as a renegade Time Lady looking for the Doctor again?
4) Well duh... we all know that a Time Lord/Lady can die if killed too fast (cf. Turn Left) without given a chance to regenerate. Also, how do we know River Song isn't Romana's 13th incarnation? The Time War went on for all the Time Lords incarnation after incarnation after incarnation.
Arguments for Romana (in addition to the arguments above)
My arguments do not stand separately, but must be considered as a whole.
1) River is exceptional. She knows a lot about many things scientific, historically, even bestially, just like the Doctor, and perhaps like other Time Lords. I'd even venture to say that she knows more than Harkness.
2) Both Romana and River Song know how to pilot a TARDIS better than the Doctor (see 4th Doctor and 11th Doctor episodes)
3) Romana and the Doctor were really close; she is one of only two Gallifreyans to have been a companion (the other being the Doctor's granddaughter Susan)
4) Romana and River both like bugging the Doctor in small ways. I think...
5) Face of Boe said the Doctor's not alone. What if he wasn't just referring to the Master?
The title is yet another pun.
If you didn't get it,
Romana + River = Roman River (two notable Roman rivers: Po and Tiber)