Benjen Stark is a bit of a lateral move in the Stark family tree, but he's an important member of the family. As the youngest child of Lord Rickard and Lady Lyarra Stark, he attended the crucial Harrenhal tournament that led to Robert's Rebellion with his older siblings. There, he and Lyanna rescued Howland Reed from an attack, establishing a loyal bond between their houses that would pay dividends during the war.
This was also the feast at which Benjen first got it into his head that he should join the Night's Watch. While he served alongside his older brother during Robert's Rebellion, fighting to avenge the rest of their family, he did not stay long in Winterfell afterward. Once Ned returned north, Benjen left for the Wall and Castle Black, eventually rising to the esteemed Night's Watch rank of First Ranger.
Of course, Benjen often returned to Winterfell. During one visit for a feast honoring King Robert — the gathering at the very beginning of "Game of Thrones" — he speaks with Jon Snow about joining the Night's Watch himself. Though he discourages Jon from joining at first, he later aids him in donning the black robes. During a mission north of the Wall, Benjen goes missing and is presumed dead. However, later on in "Game of Thrones," he's revealed to have survived, helping Bran and Jon several times before dying sacrificially. This is different from the books, in which Benjen has not resurfaced.