They could go a number of different ways with Leia's character, and what they'll do with her is one of my biggest questions about the new films.
In ROTJ, she claims she's always known that Luke was her brother, and that his power is one she couldn't ever understand or have. So perhaps Leia had given some prior thought to whether or not she could be a Jedi and concluded she could not. Then again, since she grew up in an era without Jedi, she might not have had much of an understanding of the Force, and just wrote it off as irrelevant to her life.
But of course, if Leia so chose, she could have easily been a Jedi. I doubt they'll bring up midi-chlorians in the sequels, but if Leia were to have a blood test, I'm sure she'd be chock-full of 'em – as Luke said, the Force runs strong in his family, and she's the daughter of a man who was basically conceived by the Force. Luke knew that she had the raw ability to connect with the Force, and Yoda urged him to pass on what he had learned, even mentioning the other Skywalker immediately thereafter.
So Leia doesn't think she can be a Jedi, but Luke thinks she can. If this were the pre-Empire Republic, it wouldn't even be an issue, as she'd be too old (also Yoda's concern about Luke in ESB), and her relationship with Han would be forbidden due to the Order's rules against attachment and possession. When rebuilding the Jedi, Luke will likely change those rules somehow (it doesn't matter what he did or didn't do in the EU) but from the standpoint that it was his love for his father that helped restore balance to the Force, not the fact that his sister wants to be a Jedi but also really likes her boyfriend. This is something that I feel needs to be addressed in the sequels, and not just brushed aside for people who didn't like the prequels.
At the very least, the next trilogy will have to fill us in on the backstory highlights that occurred since ROTJ. In the gap between ROTS and ANH, we knew that Anakin had kids named Luke and Leia, Palpatine had become Emperor, and Obi-Wan and Yoda went into hiding – obviously it was made later, but ROTS sets up all of those elements. Then the opening scroll of ANH told us that a Rebellion had sprung up, only recently having won its first victory, so watching them in numeric order gives you all the pertinent information. Here we really don't know what will have happened between VI and VII, as ROTJ doesn't set up anything nearly as specific.