I haven't looked for work in awhile... But I'd leave it blank if you have legally changed your name, id, bank accounts etc. If something comes up and they are serious about hiring you... They will most likely call you to get an explanation or clarification.
I was hiring a research scientist last year... And my company runs background and credit checks on all prospective applicants. Well, the guy I was interested in showed up having a felony conviction... Even though he checked the no convictions box. Mind you this is a young guy, right out of grad school with a PhD from Duke in Biochemistry... Has to be mistake right? Well I called him to discuss it, and he admitted to doing something stupid a few years ago. He said all charges were dropped, so thats why he didn't mention it. I thought for sure it had to be a mistake, but turns out it wasn't. Stuff happens. In most cases if you get far enough in to the hiring process, and they are serious about hiring you, I'd expect a call so that you can explain the name issue.