Let's start with facts that we know. R abducts L, igniting Robert's Rebellion. Eddard "Ned" Stark leaves for war during the rebellion and at its conclusion, returns to Winterfell with a baby boy, Jon Snow. He told his wife Catelyn Stark that the baby is "of my blood and that's all you need to know". He never really tells her that Jon is his son. He repeatedly thinks back to a promise he made to his sister (Lyanna) during the same time. Lyanna dies "in blood", possibly indicating childbirth.