a theory i think is pretty sound:
jon is the son of rhaegar targaryen and lyanna stark. it adds up nicely, chronologically, motivationally, circumstantially, and ned being ned.
jon was born during the rebellion. it began when rhaegar took lyanna to the tower of joy, where ned eventually showed up to rescue her. rhaegar's motives were unclear. but she was kept there for around a year. by the time ned got there, she was dying "in a pool of blood", but how she died is never revealed. the only people who knew the truth are all dead except for howland reed, who went to the tower of joy with ned.
it sounds as if she died in childbirth. and there are some in the novels who thought rhaegar had a thing with lyanna, him forsaking elia of dorne, and her robert baratheon. rhaegar by all accounts was an honorable, loved individual. it doesn't make sense that he kidnapped her to provoke robert baratheon out of malice. it makes more sense that there was an unspoken love dynamic. rhaegar had her stay there when she hypothetically became pregnant. ned, finding her as she's dying from childbirth, conceals the child and her cause of death, knowing full well that the baby would be the heir to throne and that he was committing treason, but choosing to honor his family by protecting his dying sister's only child instead. then he has a convenient cover story about how he knocked up a sailor's wench and felt all guilty and responsible. perhaps he did have an actual affair, and realized its potential utility.
lyanna stark has dark hair, and offhand at one point, a character mentions to jon that he looks more like his aunt lyanna than ned compared to his five trueborn children. it would account for jon's lack of resemblance to ned, and his dark hair when rhaegar had silvery-gold hair.
it makes sense that ned would be so hush hush about jon's origins and avoid talking about it with anyone. if it was a serving wench, then whatever, let the truth be known and move on. he knew how dangerous giving any hint to jon's true origins would be, and thought it better to conceal it until the time was right, if there would be a time at all.
also, there is the azor ahai parallel. rhaegar was thought to be azor ahai reborn by many. maester aemon (his great uncle) even thought so. now it seems jon is the favorite to be azor ahai by melisandre, who knows her shit when it comes to flames and light and such. and it makes sense. whenever she has visions of azor ahai, they are always obscured by snow. the parallels are uncanny.
this theory fills in all of the holes that surround the rhaegar--lyanna dynamic, the mystery surrounding her death, the mystery of jon's true origins, his prophetic link to azor ahai, etc.
jons snow is not dead, and he is jon targaryen-stark, heir to the iron throne, king of the seven kingdoms, and azor ahai reborn. that would be pretty dope. also dany gets lame and i don't want her to rule alone, though it seems aegon has beat her to the punch.