One might find an analogy useful. Can one harness citizen volunteerism in building a bridge?
Indeed, many might find a bridge more useful, and any experience in bridge building, materials acquisition, aesthetic design, or even fund raising can be used. However, to do a good job, the primary work should be the domain of those trained in the discipline of bridge-building. Even the advice of experienced but retired bridgebuilders should not be taken verbatim, but should always be considered and possibly dismissed against the situation of the present bridge being built.
Mathematics is not bridge building, but if the goal is to involve citizens (and not just their computers), then for a project to be successful, it must not try to get citizens to do things for which they are inadequately trained. There are many ways to run a computer program the wrong way; I would trust the masses to find a number of bugs in a program, but not to verify its correctness. So the citizens could participate in testing certain grapsable aspects of a theory, assuming there are parts that can be made accessible.
I would also turn to the masses for inspiration and for pedagogical testing. If I give a lecture to a group of people, I am interested in the feedback of those who did not understand it, or those who had a different perspective. I would appreciate any helpful (to me) efforts made to improve or broaden the scope of my communication, be it written or otherwise. I would also appreciate reasonable attempts at communicating a different perspective of the issue, so that I could "steal the idea" and use it elsewhere.
I have more than once had the fantasy of making an adventure-based computer game where the goal and steps to arrive it could be mapped either to a soution to an optimization problem or proof attempts at some interesting conjecture. Software design systems have evolved to the point where the fantasy is being realized, if only in early stages. That would be another avenue for non-trained participants to contribute.
Gerhard "Ask Me About System Design" Paseman, 2012.11.18