This badge, which I'd nominate as a silver, would be awarded for every 5 accepted answers a user has with no votes. There should probably be a 24-hour window after acceptance before this badge is awarded.
Why do this, you ask? Well, one of the much-belabored issues of the SO model is that rep is more easily accumulated by facile participation in shallow, accessible, popular topics, while successful handling of difficult, high-expertise-required issues often enough goes unrewarded. This badge attempts to act against that tendency, interpreting someone who writes a lot of accepted-but-not-voted-on answers as actively helping people solve "unpopular" problems — likely, in large part, problems from new, low-rep users who lack even the power to upvote a good answer to their question, and so may go ignored by more rep-oriented users.
Assuming that behavior is a desirable sort of participation, it attempts to provide recognition, and thereby incentive, for it.
Originally I was thinking of this badge in terms strictly of upvotes, but it seems appropriate to disqualify questions from counting toward this badge if they receive downvotes; the idea is to reward helpful answers that go mostly unrecognized, not to reward bad answers. So I've modified the qualification from "no upvotes" to simply "no votes".