While I appreciate and understand the official explanation theories about Benjen joining the Watch, I've never felt entirely satisfied by them.
Benjen was the "Stark in Winterfell" during the war, and when Ned returned, they were the only two remaining Starks, Brandon, Lyanna, and Lord Rickard all having died in the war.
In peacetime, it would make sense for a 3rd son of a Stark lord to take the black. The Starks take the Night's Watch seriously, meaning for them it is an honourable thing to do, not just a quick fix to get rid of an unneeded sibling. It is also a way to strengthen the bond between Starks and the Watch, thereby strengthening the power and reputation of House Stark.
Although with that said, I do not entirely agree. There is a lot of talk about this being a valid option, but there is also a lot more talk about finding good marriages as a political tool, to strengthen bonds between families. For a minor family, this might be considered a burden, but the Starks are a major house, and I feel confident that Benjen would have been a desirable match for a good many marriage alliances.
If you need examples: Ramsay Snow forcibly married Lady Hornwood in order to "inherit" their lands. Tyrion was forcibly married to Sansa Stark for the same reasons. Alys Karstark was in the same situation. Heck, even Edmure Tully found a match in one of the Frey's "excess daughters".
There is always things happening, and a son of a marriageable age is a major asset.
After the war, however, this situation is changed. There was only Ned and Benjen left. After Benjen joined the Watch, Ned was the sole Stark left in the world (besides Robb). While it seems reasonable for the youngest out of 4 siblings to join the Watch, while their father is still alive, it seems extreme when the entire house is all but extinct.
I am leaning towards it being a sign of grief. With so many loved ones having been killed, Benjen might have seen the Watch as an easy way out. Something to the effect of: "What is left for me here? Let my brother carry on the Stark house."
There is also the theory that Benjen played a part in the story of the Knight of the Laughing Tree, helping his sister Lyanna and Howland Reed defeat the three knights, as seen here. The gist of this story is that Lyanna, who was reputedly very fierce, skilled at arms and horseriding, played the part of the mystery knight, was unmasked by Rhaegar, which lead to their love affair, which by extension lead to the death of Lyanna, Brandon and Lord Rickard. In a sense, this could be the very event that started the war.
Even if one does not believe Lyanna was the mystery knight, it is not unlikely that Benjen played some part in what was to come, where Lyanna and Rhaegar disappear and war breaks out.
After the war, when Ned returns home with the news of their sister, I find it plausible that Benjen might join the Watch in grief and guilt.
One some level, it is also reasonable to assume that... (spoiler about Jon's mother)
if indeed Jon is Lyanna's and Rhaegar's son, that Benjen knew, and that him joining the Watch might be a way to open up future possibilities for Jon joining the Watch for his own protection. Although I am not too fond of that particular part.
Either way, I find that the official theory of Benjen's reasons is a bit too thin, and that there are plenty of room for speculation. I also have a feeling that Benjen's fate will be revealed eventually, that he has some part yet to play in the plot. For example, I believe GRRM has confirmed that the identity of Coldhands will be revealed (quotation needed, couldn't find one).