This problem most likely has nothing to do with EE…...but is somewhere in the server setup….
I’m not enough of a tech to completely troubleshoot….but, there are a few places you can look for hits…..
If you have logging (httpd) turned on, check the log from about the time when the errors occurred. It could be that you have a nasty script somewhere on the site (I had this happen once)......
If your logs are OK…..my advice would be to turn off all httpd logging.
Check your server messages…usually under var directory…to see if anything looks funny.
Use either phpmyadmin or one of the mysql tuneup tools (free) to see what they say about your mysql memory allocation. That is a very small amount of memory (256), however the site traffic is low…......these tools show whether a mysql setting (the cnf file) should be tweaked.
If you find that the problem seems to happen less when the server is restarted (for instance, after a week or so), you can set up a cron job to restart just the main tewo processes…....mysql and apache. This will not really interrupt a user if done right…....you could make it restart them every week, or every two days, etc.
Hope that helps….in general, this is a job for your server admin (ISP)....they should be able to look after the problem occurs and see why…