So, obviously Jon Snow isn't Ned's kid (even Sean Bean confirmed that).. but, even if he was a bastard of King Robert, he still wouldn't be able to inherit the throne, as bastards can't inherit. And even if they could, King Robert still has his brother Stannis. Who has a child as well..
But way back in the second episode, when Cersei visited Lady Cat, she said that she had a beautiful, black haired child that got sick and died, and then she never saw him again. What if he didn't die and what if Jon Snow was that child? That would mean that he would have a real claim to the throne. Then again, why would King Robert send him away? That doesn't make any sense..
But still..
There are lots of theories as to who Jon Snow's mother is. The black-haired first child of Cersei's is a show invention, I think... I'll have to look. But there are lots of people who believe that Jon Snow had a claim to the throne in some way. Let me know if you want some more info than that, I hate spoiling things for people!