I like to think Benjen joined the Night's Watch in response to the Mad King's actions after Rhaegar went into hiding with Lyanna.

Aerys arrested Brandon and his friends for conspiring against the life of the crown prince when they came to King's Landing intending to beat the shit out of Rhaegar for anything he may have done to Lyanna. Aerys then demanded the fathers of Rhaegar and his companions come to court so they can answer for the crimes of their sons. Once they arrived, Aerys has all of them executed in front of the entire court as a warning to any would be conspirators. His paranoia only intensified from here on out. Aerys then sends a raven to Jon Arryn demanding his wards, Ned and Robert, be executed at once and their heads sent to King's Landing as proof. This of course starts the rebellion and the downfall of the Targaryen dynasty, but the thing that never made sense to me is why there is no mention of Benjen in this story.

Benjen would be the acting Lord of Winterfell in the same way Bran was after Rob went to war. He wouldn't be I think Aerys did send a raven to Winterfell ordering Benjen to present himself before the King immediately. If Benjen complied, Aerys would be able to use him as a hostage to keep Ned from rallying the North to fight in the rebellion. If he refused the Mad King's demands, Aerys would then have a justified reason to send an army to the North to burn anything or anyone he saw as a threat and would also end the North's involvement in the rebellion before it even began.

Benjen knew Ned needed time to rally the North behind him, and that it made no difference whether he complied or refused the Mad King's demands since both choices stop the rebellion before it begins. I think Benjen took the Black in order to protect the people of the North from Aerys while also buying Ned much needed time to rally the houses of the North behind him to march south to war.

I'm probably way off but I don't feel like taking the black out of guilt over Lyanna is something he, or any other Stark would do. In my opinion, the defining characteristic of a Stark is their willingness sacrifice their own pride and honor so the people they care won't have to, even if it means they'll spend the rest of their lives carrying that burden.