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?
As with any backup solution you must test it yourself through the whole cycle first in a zero risk scenario.
I have successfully used titanium backup to this end many times. I combine it with dropsync so it's saved 'offsite' as soon as possible.
An interesting property of bitcoin wallets is that you can also observe that if you restore the wallet to a previous state then it will be rolled forward by the network when it connects again.
I should add also that should you lose your phone, you should immediately bring that backed up wallet file online and spend them into another wallet that you have sole control of to avoid someone who may have your phone spending them.
There is more than one bitcoin client for android. The one I use currently (Bitcoin wallet) allows you to use a trusted node, although I haven't test this and I don't know exactly what it does.
It is highly likely here will be lighter android clients in the near future. The Stratum overlay network looks like a promising direction.
I've just discovered a new lightweight client: BitCoinSpinner
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.