I'm finally getting ready to install an Android client. I believe Android Wallet is the most popular one.
How do I back up the bitcoins stored on it?
Also, is there a thin native Android client being developed?
Bitcoin Wallet for Android supports backing up your private keys to external storage (e.g. SD Card). Just select "Export/Import private keys" from the action bar overflow menu.
As with any backup solution you must test it yourself through the whole cycle first in a zero risk scenario.
You can then store this any way you wish, for example on paper in a physically secure location.