I downloaded the latest stable version of Moodle and upgraded HEDD. All seems to be working properly at this point.
ETA: The upgrade from 1.5.1 to 1.5.3+ does not seem to work properly for the wiki module. I grabbed the tarball I made in step 2 and made that the active version. The database now complains slightly about prior version numbers when entering admin mode, but the site works properly.
HEDD continues as it always has. We have upgraded Moodle periodically.
The current stable version of Moodle is now in production on the new server HEDD.
As with the testing server last semester, we are copying the Moodle data to PAX every night via rsync. The testing Moodle instance from spring 2004 is also available on the server as a reference, although we’re not maintaining it in any fashion.