You can just rename it and it'll show the new display name. Renaming through explorer will not rename the actual folder.
The directory name is Documents, if you look inside of it you'll see a hidden desktop.ini folder. Explorer is using the name from inside the desktop.ini and displaying that instead of the actual folder name.
If you look inside the desktop.ini you should see lines similar to the following:
If you remove the
LocalizedResourceName from the ini file and logoff and log back on the name will be whatever the folder name is.
I believe the simplest way to take care of this domain wide would be to write a user login script that copied over a new desktop.ini per folder you wanted to change (since the different folders have different icon resources you can't use the same for all) and drop a marker file to indicate you've processed this user. The marker file would prevent the script from running at each logon and allow the user to change it if the desire.
Example logon script (not tested):
if exist "%MARKERFILE%" goto eof
copy \\server\share\documents.ini "%USERPROFILE%\Documents\Desktop.ini" /y
copy \\server\share\music.ini "%USERPROFILE%\Music\Desktop.ini" /y
copy \\server\share\pictures.ini "%USERPROFILE%\Pictures\Desktop.ini" /y
echo > "%MARKERFILE%"