The issue seems to have solved itself, and I can now use phpMyAdmin again. For anybody else wandering into this thread, I was also seeing several other problems in addition to the phpMyAdmin issue.
In WordPress, none of my permalinks were working. These are the ones that require mod_rewrite to work.
In my http/error.log, I was seeing a number of "memory allocation" errors for downloadable files. The entry would appear whenever somebody tried to download a file on my site:
[Mon Aug 22 10:51:00 2005] [crit] client xxx.xxx.xx.xxxCannot allocate memory:
default_handler: mmap failed: /home/myname/path/to/download.zip
I had never seen these issues before, and they all appeared at the same time. The phpMyAdmin and WordPress problem lasted about the same amount of time. The memory allocation errors continued for about a day I believe.