1. Rheagar Targaryen loved Lyanna Stark
Nearly a year before her death, Lyanna Stark, sister of Ned, goes up the Tower of Joy with Rheagar Targaryen, son of the Mad King Aerys, from whom Robert Baratheon and the seven kingdoms rebelled. Robert ended up killing Rheagar.
2. At the Tower of Joy, Ned Stark found Lyanna in a pool of blood
It is believed that Ned promised Lyanna to care for her baby, essentially slurring his honor to hide him and keep him safe.
3. Aemon Targaryen seems to know something about Ned’s promise
In this conversation between Aegon Targaryen and Jon Snow, the old man hints at what Ned Stark may have promised his sister Lyanna by raising the question of honor versus duty. Jon Snow, as, I presume, majority of the viewers, saw this as more about him being Ned’s bastard son and whether such stature had any influence in him going to the Night’s Watch. It could as well possibly be a revelation.
4. Jon Snow was born during the war
The first book begins fourteen years after the war, the same age as Jon Snow when they found the direwolf
5. Ned Stark seems to like calling Jon Snow his “blood”, not exactly the same as his son
I have not read the books to definitively say whether Ned Stark has addressed Jon Snow as his son. In terms of the tv series, however, I can say that the Lord of Winterfell has only called Snow his blood. This is telling of Stark’s honorable nature; using such words to avoid lying. Of course, Ned did tell Jon that he would speak of his mother when they meet again. Sadly, that day will never come due to Ned Stark’s cruel demise.
6. He is not dead yet
Just on the basis that anyone could be dead at any moment, the fact that Jon seems to be well-preserved in the storyline does present an intrigue.