I have "Android debugging" enabled on this phone. When I was running Android 4.0.4 and connected to my computer running ADT-Eclipse, I believe there was a tab under DDMS perspective for browsing the file system. Assuming my memory serves me correct, is there some additional setting I need to enable for this to work again?
There was (and still is) such a tab. It's called File Explorer, and by default it shows up in the same part of the screen as the Threads and Heap views. If you've hidden it in the past, you can show it again from the Window menu → Show view, or by pressing Ctrl+3. (Thanks to Edgard Leal for suggesting the keyboard shortcut.)