I have a Red Hat 8.0 server running apache 2.x, sendmail and postgres services. Every so often the server locks up .... and it is always at 6:30am. Sometimes the server will go a few weeks, some times a few days, before it locks up.
I have checked cron and nothing is scheduled to run. I have checked everything in /var/log. No one is trying to access the website when it happens.

Okay, so maybe memory leak, right? I have a script that every few minutes echos the free command output to a text file. I have 1GB of RAM and the server never uses more than 500MB, EVER. I'm out of ideas.

You might be having some intermittent power issues. Are you on a UPS? And if so, is it able to completely regulate a brownout? A lot of them aren't, and if there's some major utility in the area of some type, it may be running some type of process at 6:30am that does bad things to your power.