It would appear that questions asked by users with 1 (or, say, <10) rep points get less attention and are often left unanswered (please correct me if I'm wrong on this). Perhaps people believe that (1) a new user is unlikely to accept or up vote a useful answer and (2) the only reason to offer up assistance is in the hopes of attaining more valuable rep points. Wouldn't it be nice if we just answered any question to which we believed we had an answer? Well, perhaps rep-hungry users would go for a badge instead.
Here's what I'm thinking...
Must be first answer to a user who, at the time, has < N rep points for some fixed N < 25 (before the voting rights begin).
Question must have been unanswered for at least minutes for some fixed T (possibly 0?).
Badge earned after M such answers for some M > 25 (I picked this for symmetry, but the point is to reward consistent behavior) for some fixed M.
I suppose this would need to be a bronze badge of which there are plenty, but it does seem to me that questions from newbies tend to get far less attention.
REVISION: I really like Pearsonartphoto's suggestion. Perhaps a better idea is to earn a badge if you edit a newbie post with no answers and it subsequently gets answered not by you. This also incorporates Ian Ringrose's suggestion that editing a newbie question may be equally or even more helpful than answering it.