I had that same problem went to an apple store (twice) and they gave me a new phone. The mail app now launches, but when I add my gmail account (both via gmail and exchange) my mail app goes crazy. It changes between different mail accounts by itself and sometimes when I go into an mail acount I just get a black screen.
I gave up anf just added the gmail app. It works well.
It happenned the same to me and I have and iPhone 4S. I tried all the suggestions many people suggested before and nothing changed. The only thing missed to make it was restore the iphone with the hope I had the same iPhone I bought (I mean with the original iOS).
However for my surprise I got the iPhone I bought but with iOS 6.1.3!!!!!!!!! OMG I couldn't believe it, but when i set up my GMail e-mail accounts all was fine, everything ran OK.
The only bad thing about this was the time I spent making backups of my pictures, files associated with some apps, doing "print screen" to each screen of my apps and the worst was restoring all these pictures, files and the apps in the same order in my restoring iPhone..... in total 6 hours........ dissapointing but I solved the problem.
I hope this solution helps to all of you.