I love working with students... as long as they work back. The goal of independent research projects is to expose students to a research domain as well as to solve some interesting problem and/or build a useful system. Ideally, we'd be able to publish some interesting result. I do not care about your background as long as you are able to put at least 8 hours of work per week on a project and meet once a week with written results.
I also prefer that students come to me at the beginning of the semester if interested.
Besides personal satisfaction and edification, and an occasional cup of coffee, independent research projects are generally unpaid.