Statistics is primarily a database oriented task. I would focus exclusively on the database first and work your way out to other systems as you learn more.
A couple suggestions:
- Run statistics during a well known downtime (overnight, weekends, etc) or with less frequency
- When the problem presents itself, log into the database and do a listing of all the database processes (within the database). Statistics will probably be the longest running process. If you see something that has been running longer, that could be conflicting with stats causing your slowness.
- Monitor your database server looking at the 4 main areas (cpu, mem, network, disk). Real time monitoring is not enough. You need to monitor and measure your performance using a monitoring tool to know what is "normal" and what is not normal.
- Upgrade your database if not running something fairly recent. Typically db upgrades come with performance enhancements.
- Upgrade moodle to 2.x
- Trim data from your database and optimize the database.