How to move and rename a project in android studio?

In Android studio 0.8.+, you can click F6 to move it.

If you use 1.0.2 or higher, make sure you select the Project view for the Refactor -> Move menu option to become available.

Thanks to @DavidsAlias for pointing this out.

Move project files as you like and then:

Tools -> Android -> Sync Project with Gradle Files

Close the project in Android Studio (File -> Close Project). Move the project and rename it by moving its home folder/directory. Open the new project from Android Studio (File -> Open)