There's not much more to say — it's a compelling theory and most people involved with the fandom take it as fact — but one sharp-minded fan came up with an additional detail that further grounds the theory of Jon Snow's true parentage.
Reddit user Dirty_Nikhil posits that all of Eddard Stark's sons are named after prominent men in his life:
Robb for Robert Baratheon...HBO
Bran for Ned's brother Brandon...HBO
...and Rickon for Ned's father Rickard.HBO
Likewise, you can see how Jon Snow gets his name from Jon Arryn — the late lord of the Vale, and the man who fostered both Ned and Robert in their youth.
If we're following that Ned named each of his kids after these important men, then why did Ned name Jon after Jon Arryn? Cue the "Serial" theme, because here comes the conspiracy theory...
"Why Jon Arryn? Ned's relationship to Arryn parallels the relationship he feels with Snow. Jon Arryn raised Ned like a son even though he was not. Furthermore, when the king (Aerys) called for Ned's head, Lord Arryn raised his banners in rebellion and defied the king to save him. No doubt Ned is defying Robert by hiding the Targaryen's claim to the throne."
Interesting, right? It's widely believed that Jon Snow is actually the son of Ned's sister Lyanna, and not Ned himself. The name "Jon," in connection with Jon Arryn, speaks to the fact that Ned is raising Jon like his own child, even if he's not his own child. And the fact that Ned keeps Jon's existence a secret from Robert Baratheon is very much an act of rebellion, much like Arryn's own bravery in the face of the Mad King Aerys.
The meaning behind Jon's name isn't exactly breaking news, but it adds another level of nuance to what many people in the "Thrones" and "A Song of Ice and Fire" fandom consider to be the most important "secret" of the entire series.
Now, we just have to wait for the official reveal on the show, and in the eventual sixth and seventh books — whenever the Seven Hells they get here! Womp womp. Insert sad Jon Snow here: