To give a viewpoint from the other side, I am an undergraduate B.S. student in Engineering, and spent the last year on a research in the Biomedical Engineering area, in Brazil.

It was based on a hot-topic of the area, and my part on the research was to verify the usability of a method invented by my mentor. So, to answer your first question, working on open-problems is very interesting to a student, as it gives you the opportunity to do something no one has ever done before, even if it gives no relevant results.

As it was a research in which I could do most work at home (programming, reading articles, etc.), my mentor and I would encounter only once a week. Each week my mentor would give me some guidance as "what to do this week", and see what I have done in the past week. We would also discuss through e-mail, when necessary. This would make sure that I was progressing in the research, and if I was not, we would discuss what should we do as an alternative.

So, answering your second question with my own experience, it was very important a regular meeting, with a goal between two of them, so that it would progress on a regular basis. More than once in a week probably would result in many goals not achieved, (which is demotivating) and less than once a week would be a very slow-paced research, with less guidance (also demotivating). So once a week worked really fine for us.

Also, you should always ask the student what he thinks could be done to solve the next step of the problem. If you don't agree with him, try to make him explain why it is a good idea. If you're still not convinced, say to them why you think that won't work, and maybe give another way to proceed the research (stating things like "I strongly suggest you do this next, instead, because of (...)"), but let him choose what should be done next, as it's his research.

Also, leave all the learning to the student. He is learning how to make research, not how to do something. You could recommend some textbooks and/or papers to read of the related theme, but it is not your job to tell him what the books/papers states. Also, it is a good idea to ask the student to give an presentation of what the papers are stating, on some of the meetings. This way, he would also be learning how to learn, instead of just learning a tool to make progress with his research. This will be very important when he starts his Master's.

This was how I was mentored. This made the research both interesting and fulfilling to me, as an scholar, and resulted in a published paper on the biggest international conference in Biomedical Engineering.

Just to be clear, again, I'm only stating my own experience as a mentored undergraduate student, that worked well for both me and my mentor. I hope that helps!