The key to nailing Edge's sound is not one thing (if only), its the total package.
It's the Vox amps pushing the high end of the spectrum. He's not into boosting the lows and mids, he goes for the highs.
It's his collection of vintage and high end stomp boxes. Overall Edge doesn't seem to be using amp distortion much, using pedals mostly instead. Although using some pedals to push the amps to distort is highly possible. Pedals Edge prefers to use are overdrives and fuzz. Ibanez TS-9, Boss OD-2, Electro Harmonix Big Muff New York version and then a slew of very obscure vintage pedals. And in case your wondering, the wah you hear over mysterious ways is a reverse wah. The sweep of a normal wah goes from down to up, that one goes up to down, creating that sticky sound.
Delays are all rackmounted units you can no longer get except on EvilBay. Overall Edge uses delay that is in sync with the beats per minutes of U2 songs, creating a cascade of reinforcing notes.
This site can tell you more about it, but use it as a guide, not gospel:
http://www.amnesta.net/edge_delay/
Delay pedals that come with tap tempo can approximate this, although there is no beating the rackmounted units. Edge prefers to use modulation delay. This means a delay where some amount of chorus or flange is added to the repeats only.
And there is the elusive shimmer. That gorgeous keyboard like sound that accompanies his guitar playing in so many songs. For which you need to split your guitar signal in the beginning, one end goes to your regular guitar chain, the other to a separate effects chain and amp dedicated to producing that shimmer sound.
You sure you still want to do this? It takes a lot of effort and money nailing Edge's sound. All the good U2 tribute bands have guitarists who use shitloads of gear to emulate his sound. You might just be better off getting a Pod XT or X3 for the odd U2 song and focus on creating your own sound instead.
Or.... don't try to emulate his sound, but if you're gonna do a U2 cover, give it the extreme make over and do something different with it instead? I once saw a youtube video of a guy who played With or Without you all by himself using a sampler/loop station and I think it was more awesome then most bedroom musicians uploading their perfect covers.