I don't have a standard user account, so I can only guess. But I can open Backup and Restore without elevated privileges. Once it is open, it will normally backup files from all users, which a standard user should not be able to do.
You can customize the backups to include just one user's files. You might check to see if that would allow for a specific user to backup their files only.
Or setup just a scheduled backup for the entire system so they do not have to worry about it.