1. Arya and Jon are described as both having the traditional “Stark” appearance; Ned’s other children look more like the Tullys. When talking to Arya, Ned mentions that she looks a lot like Lyanna did at her age. So Jon looks like Arya who looks like Lyanna.
2. When Ned finally tracked down Lyanna (who had been kidnapped by Rhaegar), he had to kill multiple members of the Kingsguard who were guarding her. Only Ned and Howland Reed survived the fight. After the fight, Ned found Lyanna feverish and dying in a “bed of blood.” She could have been dying from compilations following childbirth.
3. Ned tracked down Lyanna and fought the Kingsguard after the war. Aerys and Rhaegar were dead at that point and Robert was on the Iron Throne. Why was the Kingsguard still guarding Lyanna? The presence of the Kingsguard could be taken as a sign they were guarding a Targaryen heir (i.e. the pregnant Lyanna).
4. The dying Lyanna kept demanding that Ned swear a promise to her. “Promise me, Ned.” If she just gave birth to a Targaryen heir and that fact became public knowledge, the child would almost certainly be killed. Lyanna may have wanted Ned to promise never to reveal Jon’s true parentage.
5. Lyanna is strongly associated with blue winter roses. When Ned finds her dying, she is surrounded by blue roses. One of the visions Dany sees in the house of the Undying is: “A blue flower grew from a chink in a wall of ice, and filled the air with sweetness.” The wall of ice is the Wall, and the blue flower links Jon to Lyanna.
6. At one point in book 1, Ned is thinking of his children. He doesn't include Jon on his mental list. That could be a sign that Ned is not Jon's father.
7. Ned was a man who did the right thing regardless of the cost. It eventually cost him his life. Can it be believed that such a man not only fathered a bastard, but steadfastly refused to tell his wife who Jon’s mother was? Ned refused to tell Cat about Jon even though it was a hamper on his family and marriage. The only thing we know of that could have driven a man like Ned Stark to do that sort of thing is the promise Lyanna demanded he make.