Here's my idea...

A trusted 3rd party (like Mt Gox) would generate a block to be hashed, and send the part to be hashed to a whole bunch of different users, who generate hashes for the block.

If a winning one is found, it would be sent back to the server, which would claim the block.  The BTC would be distributed among users based on how much computation they do.

The 3rd party would of course take a cut for providing the service.

I don't know if it would be possible for the users to "cheat" by either taking the "winning" hash and claiming the block themselves, or by pretending to run hashes while actually not doing so.  The latter could be solved by making the client send the lowest hash from each batch, and occasionally double-checking and kicking out any cheaters.

So, would it be possible for a client to do hashes without having sufficient information to claim the block when a winning hash is found?

Of course the 3rd party could cheat, but if a user generated winning hash which ended up somewhere else besides the "pool", the user could find out and warn others that the 3rd party is cheating.

So here's an obvious observation. Instead of complicated pooled mining solutions, it would be possible to lower the difficulty and proportionately reduce the number of coins awarded for each generated block.

Currently the difficulty is set such that the target is 6 blocks per hour. Each block awards 50 coins. That's 300 coins per hour distributed to 6 individuals. Suppose the target were 60 blocks and the award were 5 coins per block. It would still produce 300 coins, but they would go to 60 individuals. On average, a given machine would generate 10 times as many blocks but earn exactly the same number of coins. The average time that a new bitcoin user would have to work before generating their first coins would be one tenth what it is now.

It could go further of course. The target could be 300 blocks per hour and each block award 1 coin.

Of course, blocks don't exist just to distribute coins. There may be excellent technical reasons which I'm not aware of which make it undesirable to increase the rate of block generation.

Increasing the rate of block generation would increase the number of chain forks, which makes the system less reliable and easier to attack. Focus is shifted from CPU power to network power. 10 minutes already causes a significant number of chain forks: