I use a product called SECOND COPY. THIRTY BUCKS. But I don't think it's been updated lately. It works great and it does what you're asking. It's simple and lite. See what you think.
You could do that same copy without mapping the drive, just target it as a path.. alternatively, you could use something like Second Copy ( http://www.secondcopy.com/ ) if you don't want to do this on your own.
Many usb hard drives come with backup software, and you just set it up with a nightly schedule, and leave the drive connected.
I have some family members that use http://www.secondcopy.com/, it does file-based copying on a schedule or on shutdown, with optional versioning. You could schedule it to do a copy at lunchtime, and evenings or at shutdown.
Then all you have to do is train them to leave the drive turned on, and not fill it with crap.