That Jon is not in fact Ned's bastard at all, and that the entire war to usurp the throne from the Targaryens was because Lyanna ran off with Rhaegar, breaking her betrothal to Robert, whose father then took his closest allies to King's Landing to protest her 'kidnapping' and were imprisoned, tortured and killed, eventually building into the civil war that put Robert on the throne.
Ned found Lyanna soon after the defeat of Rhaegar and her giving birth to Jon, claimed him for his own when she died, and spoiled his good name and honor(the thing he seemed to care about above all things) to protect him.
Assuming they married before his birth, this would make Jon the rightful Targaryen heir, and the likely future king.
How do you work that into the story in any other way than flashbacks? From reading the Pilot script, I really don't think they're that skilled with dialogue.