Tabs are a great starting point since they're so close to your thumb, but that's not enough. And Dropbox can afford the best.
I haven't put much thought into designing a file system app, so here goes nothing:
* The Camera Upload tab assumes that you're using it. Make it a list item in the main content tab.
* Create a special folder called Clipboard, give it its own tab with a big plus button. When you add a file to the Clipboard, you can instantly copy the link to it.
* The current sharing screen is daunting [http://d.pr/i/8v2r], and doesn't keep up with the SDK [http://d.pr/i/5Z1g].
* On Mac, let people add files to Clipboard by dragging files to the menu bar icon like getcloudapp.com.
* As a bonus, I'd like a visualization of your largest files and folders vs. those that are are accessed most often.