Right, Group Policy Preferences weren't around on 2003...started with 2008.
Ultimately, the setting is a reg key. It would be under HKCU\Software\Microsoft\Windows\CurrentVersion\Explorer\Advanced
HideFileExt REG_DWORD value of 0 to view file extensions and 1 to hide them.
You could probably use a log on script to change the value when he logs in, but like you I would probably be digging to find out why those settings aren't saving.
If your predecessor built and configured the environment, and didn't document what was done, then tracking it down might not be easy.
We still have a 2003 terminal server in my environment (that gets very little use at this point), and I logged in with a regular user (non-admin) account and was able to change those settings and they were retained after logging off/on...so It's not necessarily something inherent in users not having the privileges to save those changes on a 2003 TS. But it's been so long since I've done any real work in a TS environment that I can't recommend anywhere else to look...