Can I use Apple TV First Gen as a wifi extender?
It should work if you do as you suggest - hack your Apple TV to install OS X on it, you should be able to use Internet Sharing to share the wifi signal. There are many resources on how to do this, but this blog post should be a good starting point.

Two caveats:


I haven't tried this, so no promises.
Because of how Internet Sharing works (it's using NAT), whatever is on the wifi side of the network may have limited access to the rest of your network. Internet access should work okay, but things like iTunes Home Sharing and the like won't be accessible.