If I recall correctly, Ned kinda asked him to. When he was walking to his death, he passed through Yoren and told him "Baelor" (video here), referring to the statue of Baelor, where Arya was. He knew it was Arya because they met previously when Yoren arrived King's Landing and went to see Eddard at his room, where, once again, Arya was.
Why didn't he ignore it? Well, it's something a sworn brother of the Night's Watch just couldn't do (gladly)
Know that feel bro...
I also think his chapters are way much boring (IMO) than the rest of the story, which is quite thrilling, and that might be a reason.
It might also relate to the fact that I'm sorry for his legs, and associate his chapters with some kind of depression I dislike while reading a story like this.
I also agree Jon had it all figured out. As soon as Melisandre wanted to kill the baby, Jon would reveal his identity (even though he hasn't a name ) and then the sacrifice would be nonsense.
Of course it's not a popular decision, but it certainly was one of the most clever he has made (and will probably ever make)