Based on your initial description my first thought was that the student had cookies disabled, which would prevent the browser from maintaining a session when moving between WeBWorK pages, but the fact that this happens with different browsers makes me doubt that.
In terms of the student showing as "Active", all that means is that the student has a valid WeBWorK session. After logging in, WeBWorK writes session information to the database which is kept for an hour, so as soon as the student logs in they will show as "Active" for an hour.
I'm a little surprised that rebooting the server fixed this. It sounds from your description like the student's computer was doing something weird, but the fact that it happened for multiple browsers is hard to explain. Each time the student logs in it creates a new session, so I would expect the second login to clean up any corrupted/stale session.