Before listing some concrete problems -- that I think can be a nice student project-- I would like to comment on our mutual expectations out of a project.
A student project depends not only on my own areas of interest, but also on the background and interests of the student. So in principle we can find a common problem of interest to work on that is quite different from what is listed here. Depending on your level the outcome of a project can vary from understanding some research level project to becoming an expert in some particular area of research. Some projects can be finished by implementing some research ideas into concrete software. (Disclaimer: Hands-on implementation work under my guidance is recommended only for bachelor projects. Even these will require you to read and understand theoretical results!)
Independent of the scale and final outcome of a project that we may mutually agree upon, it is expected that at the end of the project you have learnt about methods and objects that are of theoretical as well as practical importance. In general, they will relate to geometry and polyhedra.
Here are some concrete problems for projects.