Just to update, that worked perfectly. And to reiterate, here's what I did:
1. Connect to the Mirror DB Server, open SSMS
2. Execute this script to find the file names:
SELECT name, physical_name
WHERE database_id = DB_ID('<dbname>');
3. Execute this script on the Mirror DB
ALTER DATABASE <dbname> MODIFY FILE (NAME = <file>, FILENAME = '<newpath>')
I get a notification:
The file "<file>" has been modified in the system catalog. The new path will be used the next time the database is started.
4. Stop the Mirror Server services.
5. Copy the files to the new locations.
6. Start the Mirror Server services.
7. Wait for everything to sync up again.
8. Delete the old files.