The Benjen/"Coldhands" connection
In the books, the fate of Benjen is still unknown. When he returned again in Season 6 in his infected form, some readers of the Song of Ice and Fire novels thought that the character was an adapted version of the novel’s “Coldhands” character, a mysterious ranger who rides an elk and helps Bran on his journey towards the Three-Eyed Raven (in the fifth novel A Dance With Dragons), plus saves Sam and Gilly from Craster’s Keep (in the third novel, A Storm of Swords).
George R.R. Martin has explicitly stated that in his narrative, Benjen is not the mysterious “Coldhands," but in a Behind the Scenes feature on Benjen's Season 6 return, creator D.B. Weiss even refers to him as “Coldhands,” even though that moniker never made it into the dialogue of the show. The differences between the two have only increased with the scant few episodes we’ve had featuring Benjen/Coldhands.
The show replaced Coldhands's storyline featuring Sam and Gilly with Sam learning how to use a dragonglass dagger and summoning the courage to trek out on his own. The killing of Night’s Watch mutineers at Craster’s Keep was given to Jon Snow, who needed something to do for a few episodes that season. Jojen and Meera Reed absorbed the task of getting Bran to the Weirwood tree cave and meeting the Three-Eyed Raven. It wasn’t until after Bran had taken on the Three-Eyed Raven responsibilities that Benjen showed up, claiming to have been told by the previous Three-Eyed Raven that he needed to protect Bran.
Presumably, Benjen didn't find them at the Weirwood Tree Cave because the same magic that held the Night King outside (until Bran messed it up) would also keep Benjen away. On the show (not in the books), Benjen tells Bran that he was attacked by White Walkers and transformed into a wight when the Children of the Forest stepped in and shoved dragonglass into his heart, stopping the wight process but making him technically undead. This is also something we saw happen to the Night King in one of Bran’s visions. Which, in a way, complicates the Night King’s character. Benjen keeps some semblance of his personality in his transition to undead, which suggests the Night King isn’t a mindless force of winter, but a character that could (potentially) have motivations to punish both the Children of the Forest and the Realms of Men.
Benjen sticks around with Bran long enough to deliver him an ominous-yet-encouraging message about the impending war with The Night King: "One way or another, he will find his way to the world of men. And when he does, you will be there waiting for him. And you will be ready."