Exactly what I'm thinking. Technically speaking, that is what a video card, processor, or FPGA does, just at inhuman speeds. What I had in mind was more of a planned approach. 'Man, x hasn't been used, let's try that!'
I'd start work on it, but would it be as simple as changing a mining program, or bitcoind or something?
I don't think you understand how mining works.
You aren't looking for a random hash you are looking for a nonce which causes the block header hash to be less than the target.
Sure you could write a miner which returns random garbage to a pool. They (all) will promptly ban you as your reject rate will essentially be 100%.
A hash by itself is useful. It isn't like you hand over a hash 123....999 to the network and they give you a block reward.
You hand over a COMPLETE BLOCK HEADER with a hash meeting the target and the block is accepted by the network. Random hashes of nothing are worthless. Sure eventually you will find one which is small enough but it is STILL WORTHLESS. It doesn't correspond to any valid block header.