Can Bran sire children?
House Stark is not doomed. Bran is the only surviving legitimate male Stark but he is probably unlikely to father children due to his disability. But he is not the only Stark alive.
Since some of our members seem to believe that Bran can have intercourse and that medical science in Planetos and our world are same, I'd like to quote Eddard Stark:
“I dreamed of Bran,” Sansa had whispered to him. “I saw him smiling.”
“He was going to be a knight,” Arya was saying now. “A knight of the
Kingsguard. Can he still be a knight?”
“No,” Ned said. He saw no use in lying to her. “Yet someday he may be
the lord of a great holdfast and sit on the king’s council. He might
raise castles like Brandon the Builder, or sail a ship across the
Sunset Sea, or enter your mother’s Faith and become the High Septon.”
But he will never run beside his wolf again, he thought with a sadness
too deep for words, or lie with a woman, or hold his own son in his
That is the only mention of Bran's ability to procreate as far as I can recall. And his own father seems to believe that Bran can never lie with a woman aka have sex and he would never sire a child.
But I also have to add that I am not a medical expert to provide a definite ruling on the matter. I am only building on what the author has provided us with in the books.
In any case, Can Bran have children? Maybe, Maybe not. Do other characters in universe seem to think he will have children? No they don't.
Other surviving Starks
But despite all that, Bran has two legitimate sisters; Sansa Stark and Arya Stark. Arya is still too young but Sansa is of age to bear children. Any children Arya and Sansa may bear can carry the Stark name and inherit the North from their uncle Brandon Stark. Any man they might marry can also take Stark name and the children born out of such union will be just as much Stark as children of a male Stark.
Also in the books, Rickon is still alive and is therefore heir to Bran.
Precedence for female line
There is precedence for female line furthering a great House.
This is not the first time for House Stark to look to female line for survival if Wildlings can be believed. Lord Brandon Stark fathered no children except a daughter. The daughter was supposedly abducted by King Beyond the Wall, Bael the Bard. Brandon was distraught because this meant extinction of House Stark. But later the girl was found with a child fathered by Bael. That child was brought up as a Stark and succeeded his grandfather upon his death.
House Lannister was once in a similar dilemma. King Gerold III Lannister died without a male issue. He had only one daughter and she was married to a Knight named Joffrey Lydden. Upon King Gerold's death, Ser Joffrey was crowned as King of the Rock by council of Westerlander nobility and he took the name Lannister due to his marriage to the Princess of the Rock. He ascended to the throne as King Joffrey I Lannister. Current Lannisters are descended from King Joffrey and daughter of King Gerold, no one considers them to be "not real Lannisters".
Another example is given in succession crisis of House Hornwood*. Lord Halys Hornwood died in battle against Tywin Lannister. His only legitimate child Daryn Hornwood died in battle against Jaime Lannister. The considered candidates to carry on Hornwood names were:
- Larence Snow - A bastard son of Lord Halys Hornwood. He could be legitimized by King in the North to be the next Lord Hornwood.
- Beren Tallhart - Younger son of Lord Halys' sister, Berena Hornwood. His father offered that Beren would take Hornwood name if he was to be named heir of Halys Hornwood.
Another example is kindly reminded by Broklynite. Lady Maege Mormont's children carry the Mormont name. It is unclear who is the father of those kids. When Maege's brother Jeor Mormont abdicated his seat and took the black, Jeor's son Jorah Mormont became Lord of Bear Island. Jorah then went into exile without siring any kids so Maege Mormont became lady of Bear Island. Her children are considered Mormonts. (But this would be a weak example because we do not know who is her husband. If he is/was a Mormont as well, this example won't hold).
So even if Lords of the North refuse to have Sansa as their Queen like Lords of Westerlands refused to have King Gerold's daughter as their Queen, Sansa's husband can instead be crowned as King in the North (Now you know why Petyr Baelish was so eager to marry Sansa) if he takes the Stark name just like Joffrey Lydden.
* The Succession crisis of House Hornwood never occurred in the show.
Who can take Stark name?
By law, a man can only take Stark name:
- By right of marriage to a Stark woman
- By right of descent from a Stark woman
- By grant of legitimization decree for removal of bastardy taint from the King provided he is a bastard of a Stark man or woman.
- By right of birth to a Stark man in wedlock (The normal way)
There is no other way or precedent to be a Stark. So Arya, Jon and Sansa all have equal potential to further the Stark line even if Bran cannot.
Why show hasn't showed that potential succession crisis?
Because there is no succession crisis. if Bran comes back to take his place as King in the North as eldest surviving son of Eddard Stark and reveals that Jon is not a Stark, House Stark is still not doomed. Even if Bran fathers no children, Sansa stands as his heir. Whoever marries Sansa can take the Stark name for himself and the children of that man and Sansa will be named Starks. Even Jon can take Stark name if he wishes due to his descent from sister of Lord Eddard Stark if Bran legitimizes him (Ironically, Jon would then also be able to take Targaryen name, not just Stark name).