Xna could be used to develop a miner in C#. It is the only legitimate way of running homebrew on an Xbox. You need a developer subscription to deploy software to the console but they can be obtained for free on Dreamspark.
Forget about using the GPU - it is more like an enhanced version of an ATI X1900. Gpgpu functionnality would have to be emulated with pixel shaders - good luck coding a sha256 implementation in ps3.0
The CPU packs quite some horsepower for its age but I would expect it to perform below the PS3's Cell - probably around 10 to 15 mhash/s at best. I don't know how much overhead the xna .net wrapper introduces compared to native PowerPC code. The xbox power consumption is quite big (200w for the xenons !), older models would probably overheat /rrod if left on mining for days.
Networking is also very restricted under XNA but debug builds allow for unencrypted communications on the LAN - so as long as you are running a proxy on a computer (or mining solo) you should be fine.