I believe Edge was named as such for his sharp wit (as noted above) and also because the young Dave Evans was a quiet guy who was always sort of "on the edge" of the Lypton Village crew (the name Bono & his group of high school friends gave themselves).
The names Bono and The Edge both came from Lypton Village. Bono (in full Bonovox) was the name of a hearing-aid store and The Edge came from the guitar sounds he produced.
Like many before me have already said: "because of his sharp mind". And I'm not sure if it was Bono who named him - I always thought it was just 'the Lypton Village guys'.
Bono gave The Edge his nickname, but he's a bit cryptic about why. When he's asked, he grasps Edge's long, chiseled jaw and turns it in profile: "The Edge." Then, after a pause: "Let's just say he's on the border between something and nothing."
In the VHI documentary, "Legends", The Edge mentioned that when they were growing up as kids, friends gave each other nicknames that "nailed their insecurities" and he mentioned it was because of his chin.
So who had this preposterous idea to call him The Edge?
I do preposterous in this band. It had something to do with the shape of his head, his jaw, and an insane love he had for walking on the edges of very high walls, bridges or buildings.
This might sound really stupid, but I think the point is that no one really know exactly why he's called the Edge, since everyone who called him that had their own reasons. And I also think it's something Bono enjoys changing the answer to, since the question pops up in interviews every time they are releasing an album. But maybe I'm wrong. I've heard it's because of his chin most times anyway...