A daemon that writes the contents of the clipboard to a log file would be a reasonable utility. It does not need to be pauseable; just kill the process when you don't need it. It doesn't need to only record URLs; just use grep to filter the log file. This is how I would write the thing for my own use.

The professor gave the students incomplete information. What they said is perfectly reasonable. This is a project they will be doing on their own time, so the scope of it is entirely up to them.