I offer this answer with a disclaimer. I am an entering graduate student and have no experience mentoring undergraduate research, although I did do some undergrad research.

I can't tell you what to research, or how often to meet, or how to specifically guide, but I can tell you, what is in my opinion, the right mindset. Very few undergraduate research projects result in important results (I'd be curious to see an example of one though!). The point of undergraduate research, in my opinion, is training. It is to teach a young mathematician the tools they will need to succeed in whatever path they are currently intending; industry, research in pure or applied math, etc.

I'll explain my undergrad research, which if I could be frank is not research in the sense that I solved an open problem; my senior thesis was entirely expository. I investigated the relationship between Brownian motion and PDEs like the heat and Schroedinger equations.

During my investigations, my adviser taught me a lot of various areas. I learned PDEs, probability, stochastic processes, Fourier analysis, functional analysis, quantum mechanics, statistical mechanics, etc. Sometimes he would just teach me things unrelated to the project, just because I asked a question and/or he thought it was important.

What I think is important is learning the tools of the trade, whatever the student needs to succeed in their intended career path. I was introduced to a really interesting and exciting topic, which was bonus. Currently I am intending on researching SPDEs and rough path theory. This may change, but no matter what, as a researcher in math I need to know functional analysis, etc. if nothing more than to pass my quals (studying for these have been considerably easier because of my project).

tl;dr The topic itself matters less than the tools you teach. It is more important to teach the necessary tools to succeed in research than to actually produce real results.