It's more complex than it ought to be, but this procedure worked for more than just the Dropbox SDK.
- Unzip the Dropbox Chooser SDK somewhere
- Open your project in Android Studio
- Create a new sub-directory "libraries" (right-click -> new -> directory)
- Create a new module in the libraries directory (right-click -> new -> module)
- In the "New Module" window, from "More Modules" select "Android Library"
- Click Next
- Edit "Library name" to "dropbox-chooser"
- Edit module name to "dropbox-chooser"
- Edit package name to "com.dropbox.chooser.android"
- Click next until done
- Navigate to "libraries/dropbox-chooser/src/main"
- Delete everything within the "java" and "res" directories, but do not remove the directories
- Navigate to the directory where you have unzipped the SDK
- Copy everything in the "src" directory to the "java" directory of the new module
- Replace the "res" directory in the new module, with the one from the SDK
- Do the same with AndroidManifest.xml
Now, to reference the library in your main project:
- Open/Create "settings.gradle" in your main project dir
- Add include 'libraries:dropbox-chooser' to the file
- In build.gradle, withing the dependencies section add compile project(':libraries:dropbox-chooser')
This should do the trick.
Sync, build, and all should be good.