In a recent conversation over drinks with a colleague, we got into the classic debate of whether researchers should select interesting questions or important questions. There are lots of "research advice" documents out there discussing this tradeoff (eg. this one by Donald Davis, which DRDR recently pointed out), but I don't believe there really is a tradeoff per se.
Here's what I think my opinion is (for now, at least): there are many interesting but un-important questions but there are no important but un-interesting questions.
If a question really is important for society, then it is by definition interesting. If an important question remains unanswered, it is probably because it is difficult to answer. But if it really is important, then there will be many people interested in its answer (so long as you can come up with a good one). Clearly, this means the question is interesting.
I think there are two reasons that important questions sometimes get mistakenly categorized as "uninteresting" by researchers. First, researchers sometimes have a strategic incentive to label a question they do not know how to answer as uninteresting, since it gives them justification for not investigating the problem. Second, questions may be so difficult to answer that we assume any answer will be unsatisfying and thus uninteresting. In both these cases, I think we confuse the difficulty of a problem with whether that problem is interesting.
It's fine with me if some academics want to work on interesting but unimportant questions, I don't see any way to stop this from happening. But I get angry when researchers thoughtlessly dismiss as uninteresting a question that they are willing to recognize is important.