This is not really the use case of git. It's not a simple copy. You can look at old changes, reverse old changes, branch and merge new features at a later time.
It's does everything you would need to do manually do with copy paste (diff, merge, find old changes, comment changes, etc) but with much less effort (assuming you use a GUI).
1. Click folder in Windows Explorer
2. Press Ctrl+C
3. Press Ctrl+V
4. Press f2
5. Type new name
6. Press Return
Mac OS X:
1. Click folder in Finder
2. Press Cmd+D
3. Press Return
4. Type new name
5. Press Return
(No real idea about Linux, which I'm only confident with from a programming perspective.)
If nothing else, it's at least fewer keypresses.