I don't know who first proposed the theory that Jon is the product of a union between Rhaegar and Lyanna... but I believe it.
Fact: Lyanna is known to be with Rhaegar.
Brandon hears the news while on his way to Riverrun to marry Catelyn. From the Starks perspective (and Robert's), Lyanna was kidnapped while the Targaryen perspective seems to be that Lyanna was willing to run away with Rhaegar or that Rhaegar merely took what was his due as Crown Prince. Brandon rides to King's Landing and demands Rhaegar's head. Aerys captures/arrests (depdnding on Stark and Targaryen perspectives) Brandon and calls Lord Rickard to come answer for his son. Aerys then murders/justly condemns Rickard and Brandon. Lyanna is still held against her will/freely living somewhere with Rhaegar. Aerys then called for Jon Arryn to hand over the innocent/traitorous Eddard. Ned, Robert (Lyanna's betrothed), and Jon raise their standards in revolt rather than let Aerys murder innocents/judge traitors.
It's pretty much been proven from Jaime's POV that Aerys was a bloodthirsty lunatic. So we can agree with the Stark perspective on the actions of Aerys.
But as for Rhaegar's actions, Robert and Ned seemed to have hated him. But in Dany's POV, Barristan provides a completely opposite opinion of Rhaegar. We also see a little of this in Jaime's POV. Barristan and Jaime both viewed Rhaegar as gentle, honest, forthright, noble, courageous, and highly intelligent.
Also, from Ned's POV and from Meera's stories, we learn that Lyanna was extremely headstrong and willful. She might have run away with Rhaegar.
The theory is strongly influenced by the idea that Lyanna ran off Rhaegar instead of being kidnapped by him. Jon's birth could have been the result of rape, but I think not.
But perhaps the strongest argument for Rhaegar and Lyanna's parentage of Jon comes from Ned's memories of Lyanna and this promise he made to her. She was dying in a bed of blood, perhaps from complications in giving birth. She demanded a promise from Ned... this promise cost Ned his whole life. He thinks about how hard it's been to keep this promise over the years. If the promise was to bury Lyanna at Winterfell, it does not seem that it should be hard to keep fourteen years later. If the promise was to protect Robert, well Ned made up with Robert after the rebellion, Ned let Jon Arryn watch over him, Ned led the counter attack against Balon Greyjoy's rebellion, and Ned served Robert as Hand... it seems that Ned was just doing his duty. If the promise was to hide Rhaegar's son from Robert's hatred of Targaryens, then this seems plausible. Claiming Jon as his ******* cost Ned intimacy with Catelyn, cost him humiliation for being disloyal to his wife, cost him slight upon his sacred honor... yet he let Jon grow up with his own children.
When AGOT starts, fourteen years after Robert's ascension to the Iron Throne, Jon is fourteen years old.
There are other threads where this is spelled out in more detail...