Well actually

(no not the maths)
When you reboot a miner it gets a new unique number that's in the coinbase sig as I mentioned in that long post.
So it does actually change your hash results

Of course it's random ... but it does change it.
So, in theory, the newly generated hash might be what ends up solving a block? maybe the old one is worn out... :-) Its like playing the same lottery numbers over and over again? but I just changed the lotto numbers?
In theory, yes. On the other hand, the pool sends you new "lottery numbers" (approximately) 3 times per minute... and there's the peculiar situation that there's an extremely small but finite chance ("So you're saying there's a chance?") that the next block could have been hiding a few trillion numbers away in the nonce that was being used before you rebooted.

Rebooting would probably be more beneficial if you were worried there was some sort of hardware issue where chips or memory weren't rolling properly.