This started off as a joke in my head, but then what initially was a ridiculous theory started to make a little sense?
First off, JR Orci is a major contributor to the show, who also did Fringe, and while the show isn't in nearly the same scifi territory as Fringe, TV.com reviewers have noted that the show has taken on some Fringe-esque elements.
So, with that in mind, and with Red's confession at the end of the midseason finale that he in fact was not Lizzies father, I started thinking. It would be a bit of a lame writing convention if down the road he basically just says "Oh I actually AM your father, I just lied on the phone, sorry, not sorry."
He very well may not be her father....so what the heck is happening? Well based on what I know about how Orci writes, I think there will be a truth/half truth twist. So what do I think?
Red is either Lizzies handler, of sorts, or he is related to her genetically. Both of these theories kind of converge.
Remember the box full of random stuff in Lizzie's house incriminating her husband? Well I have a feeling that box may have actually belonged to Lizzie (as she does have a somewhat matching burn brand on her arm).
Basically, it's Lizzie, I feel, who is not who she appears to be. Maybe she was a genetically bred soldier MKUltra style (explaining both her vague past, and her strange but not direct relation to people like Red, as he probably contributed his genetics to the project, explaining his relation to her, interest in her, while still not truthfully being her dad). I also seem to recall Alan Alda implying Red had some very important thing that made him untouchable (I thought at first it was something basic, like an atomic bomb that would go off if something happened to him, but I think that too may have something to do with Lizz).
Anyway, I'll be very surprised if I am hitting the mark straight on with this, but I certainly won't be too stunned if I squeezed in some lucky guesses in there.
Alternatively, if perhaps the show ever intends to go fully off the rails, Red and Lizzie could actually be the same people, gender swapped from parallel universes .