The feature to "star" or "favorite" a problem already exists; if you go to the problemset archive, you can choose to mark any problem as favorite. Same holds for marking any contest.
Unfortunately, it seems the only use for this feature, is so one could revisit his favorite problems in the future — you can't view the favorite problems of other participants. I think this feature can be easily extended to something more meaningful.
Maintain for each problem, the number of accounts who liked it (now referred to as the likeability of the problem). Then implement the option to filter (or sort) problems by their likeability (hopefully filtering problems by likeability could be combined with sorting by the number of people solved, so that people could find their own difficulty level). Same could go for contest likeability.
I think this can mainly help beginners. I hope that people who try out competitive programming for the first time, will first get intoduced to good problems. This way, they will get intrigued to solve more problems — I know this is what's gotten me and a few friends to invest into this branch. (Of course, this means that the option to filter or sort problems by likeability should be very apparent for any newcomer)
I believe this will improve the CF problemset as a source of practice, which is probably a main intention of this problem archive.
I can't find any at the moment. Feel free to point out some in the comments.
Perhaps Mike would want to weight the likeability marks, depending on the rating of people who marked it. This is open for discussion of course, but I think Mike would have better expertise in that area, so I'd like to hear his opinion.
Though unlikely, I suppose some people could create fake accounts to like their own problems, for unknown reasons. So, it could be implemented to include only rated accounts' marks into the likeability of a problem.
Contests' likeability could increase the contribution of all authors. Just a side note though.
Everything is open for discussion. You're welcome to throw in more pros, cons or side notes.