Yesterday I proved a small fact and asked a follow-up question in this answer:

http://mathoverflow.net/questions/22585/algebras-over-the-little-disks-operad/45620#45620

It's fairly elementary, I'd be interested to know more about it, and I have never seen anything like it in the literature (although I have not searched). So you could look at that if you wanted to. More generally, there are plenty of problems like this, but they are not always easy to find. If you just start reading books and looking for things to research there is a danger that you will just be led along the best-travelled paths where enormous amounts of work have already been done. So I would second the advice to ask several professors for suggestions before deciding on a topic.