Frequently Asked Questions (from westeros.org)
How can Jon be a Targaryen if he has a burned hand?
Targaryens are not immune to fire. Aerion Brightflame died drinking wildfire. Aegon V and his son Duncan are thought to have died in a fire-related event at Summerhall. Rhaenyra was eaten by Aegon II's dragon, presumably roasted by fire before the dragon took a bite. Viserys died when he was crowned with molten gold. Dany suffered burns from the fire pit incident at the end of A Dance with Dragons. Finally, the author has stated outright that Targaryens are not immune to fire. Jon's burned hand does not mean he is ineligible to be part Targaryen. For more information about the myth of Targ fire immunity, see this thread.
How can Jon be a Targ if he doesn't have silver hair and purple eyes?
Not all Targaryens had the typical Valyrian look. Alysanne had blue eyes. Baelor Breakspear and his son(s) had the Dornish look. Many of the Great Bastards did not have Valyrian features. Jon's own half-sister Rhaenys had her mother's Dornish look.
If Jon isn't Ned's son, then why does he look so much like him?
Much is made over the fact that Arya looks like Lyanna, and Jon looks like Arya. Ned and Lyanna shared similar looks.
How can Jon be half-Targ if he has a direwolf?
Ned's trueborn children are half Stark and half Tully. Being half Tully didn't prevent them from having a direwolf so there is no reason to think being half Targaryen would prevent Jon from having a direwolf. If Lyanna is his mother, then he's still half Stark. Furthermore, there is already a character who is half Targaryen and half blood of the First Men and was a skinchanger: Bloodraven.
Since Rhaegar was already married, wouldn't Jon still be a bastard?
The evidence that Jon is probably the legitimate is that Targaryens have a history of polygamous marriages which makes it a possibility that Rhaegar had two wives. Three Kingsguards were present at the Tower of Joy when Ned arrived. Even after Ned said that Aerys, Rhaegar and Aegon were dead and Viserys had fled to Dragonstone, the KG opted to stay at the TOJ stating they were obeying their Kingsguard vow. The heart of a KG vow is to protect the king. With Aerys, Rhaegar and Aegon dead, the new king would have been Viserys, unless Lyanna's child was legitimate making him the new king of the Targaryen dynasty. For a comprehensive analysis of Jon's legitimacy, see the detailed explanations in the two linked articles.
This theory is too obvious and too many people believe it to be fact. How can it be true?
The theory is not obvious to the majority of readers. Some will get it on first read, most will not. Keep in mind that readers who go to online fan forums, such as this one, represent a very small minority of the ASOIAF readership. Also, A Game of Thrones has been out since 1996. That's more than 15 years of readers being able to piece together this mystery.
Why doesn't Ned ever think about Lyanna being Jon's mother?
Ned doesn't think about anyone being his mother. He says the name 'Wylla' to Robert, but does not actively think that Wylla is the mother. He also doesn't think of Jon as his son. There are numerous mysteries in the series, and Jon's parentage is one of those. If Ned thought about Jon being Lyanna's son, it would not be a mystery.
Why should we care who Jon's parents are? Will Jon care? Who cares if he's legitimate?But Why? (from mlarchives faq)
Once one accepts that the evidence is conclusive and that Jon's parents are Rhaegar and Lyanna and that he is most probably legitimate, these become the important questions.
The way most characters describe it, Rhaegar seems to be a completely honest and honorable man. So why would he betray his wife and run off with Lyanna to father a legitmate child?
This is a question that cannot fully be answered, except for one vision that Dany has in the House of the Undying on CoK, pg. 527. In this scene, Dany sees Rhaegar and Elia standing over their newborn son Aegon. Rhaegar says that Aegon is the ‘prince that was promised,’ and his is the ‘song of ice and fire.’ Then, he enigmatically adds, ‘There must be one more...the dragon has three heads.’ The first two heads are evidently Rhaenys and Aegon. It’s possible that in whatever prophecy Rhaegar read, he thought that he needed to father a child off a Stark to produce the third head and the ‘song of ice and fire.’So I should make a conclusion, but I won't, we all just have to watch out for the next books to determine the truth. This was something like Naruto. Many many years ago, I still remember fighting for Minato as his father. Some passionate debater said something about the lines of "Why, Sasuke looks like Orochimaru, that doesnt make him his father". I have no idea where that logic came from, but I keep on dreaming what his expression was when he realized I was right *evil laugh*