Well, I'm not sure that there's any hero who I could call "perfect", but I'd certainly agree that inner turmoil definitely makes any character much more interesting. It's rarely enjoyable to see a hero breeze through challenges without some problems to deal with, and internal hardships are all the more interesting when they're done well. I'm actually planning something along those lines myself in WDAMO, but I can't really say anything more than that at me moment. But struggling against oneself definitely makes for more drama and it often allows the reader to more closely relate to the characters.
It's not a book at all (unless you count manga adaptations), but I think the anime Neon Genesis Evangelion does an excellent job of using inner turmoil as not just a piece of the puzzle, but as a central theme. Frankly, every last one of the characters has some major problem they're dealing with, from the main hero, Shinji, who deals with an absent and uncaring father, to the very minor character, Touji, whose sister is rendered comatose in an accident (by Shinji, no less). It's very unusual, but it also makes things vastly more interesting. NGE uses emotional problems in the best manner of any dramatic work I've ever seen, to be frank. (Note to those considering viewing NGE: It's rated 16+, I believe, so viewer discretion is definitely advised.)
Other qualities that I think can be good in heroes are unlikely heroes, like Chris mentioned, and heroes that really shouldn't be heroes. The title character of the Artemis Fowl series is a conniving, ingenious young boy who is an extremely successful thief and swindler. One of his favorite pastimes, in fact, is outsmarting his teachers, psychiatrists, parents, etc. I believe he made a few of his professors break down and cry a few times, in fact. You know you should hate him, but you can't help but love him when the plot and the characters begin to develop. It's really remarkable.
Of course, these things are only good when they're done well. I've seen plenty of catastrophes when writers fail to allow their readers to relate to protagonists who are either unrealistic or unlovable. It's definitely tricky.