Here are a few guesses of areas where you could start looking.
Is swap initialized and working OK? When a machine is idle for a while swap will sometimes utilize the idle machine time to write a large process to the swap file as a head start for when things get busy. If swap is not set up you could conceivably hang.
Do you have a screensaver enabled? There is the ocassional bug that when a screensaver starts, the machine freezes.
Do you run cron jobs periodically during the day? You could have a screwed up cron job that hangs the machine every time it automatically starts.
"I can't find any clues in any of the other logs I find in /var/log"
You also might take a look at the size of the log files. Is there something that is spewing out large amounts of innocuous error messages?
Be prepared. Create a LifeBoat CD.