1) The Doctor isn’t the last Time Lord because of
2) The Master
But the Master went down with the rest of Gallifrey, and River can’t regenerate. Is Clara River Song? No, because we know for a fact that River Song is her last regeneration, because the Doctor saw her die and not regenerate. That’s a fixed point in time if ever there was one, because it forms her entire relationship.
2)The Time Lords are returning
A big season-busting twist, but already used once. The Time Lords returned through the long-established drumming in the Master’s head. Could Clara be a new anchor for them to return? Maybe. But if she’s some sort of function of Time Lord technology, why does the TARDIS dislike her so much? You’d think they’d have a special understanding.
3)Is Clara the Doctor’s Mother
I’m assuming that this is the woman from The End of Time, although I don’t know for sure. This is, creepily, a distinct posibility. What we know about Mom is that she can turn up out of nowhere and disappear out of nowhere, even though she’s supposed to be locked up with the rest of the time lords.
4) Is Clara the Doctor’s Daughter
Is Clara Jenny? Jenny vanished after one episode, and her paths haven’t crossed with the Doctor’s again. So this is possible. But Jenny is a bit of a different time lord: she doesn’t regenerate, she just doesn’t die. So it doesn’t make sense for her to now be Clara.
Personally, I find 3 and 4 both highly unlikely, because I don’t think Clara’s blood family to the Doctor. I don’t think the writers would have had her make out with the Doctor in the Christmas episode if she was his relative, and I don’t think they’d be as flirtatious as they are if the overarching plot was to have them be related.
But ultimately, I think Clara cannot be a Time Lord. Time Lords regenerate. They regenerate at different ages, and they hop around time like River Song, but the regeneration itself is still necessary for them to live again.
But Clara hasn’t regenerated. The Doctor actually watched her die once, and then buried her–she definitely didn’t regenerate. And in the Dalek episode, her body had died but she herself had continued living: she should have been able to regenerate before the Daleks assimilated her. And in “The Bells of St. John,” her body dies twice without a sign of regenerating.
She’s had three different families, three different established timelines that had absolutely nothing to do with the Doctor. Babyhood to death, she’s lived twice without regenerating. I don’t think she’s a time lord.
UPDATE: I’m going to officially rule out the idea that Clara is in any way related to the Doctor. Not Jenny. Not his and River’s daughter. I’m basing this on the tight skirt line. There’s no way the show has the Doctor making that face while thinking about his granddaughter’s butt. Nope. Not happening.