If a massively online collaboration is set up, then it should try to focus on problems with a reasonable chance of success. The three classical construction problems of antiquity are known as "squaring the circle", "trisecting an angle", and "doubling a cube". Modern mathematics resolved all three, but much more important was the earlier Descartes revolution, which enabled mathematics to free itself from the mental prison of compass and straightedge constructions. Notice that the Greeks used compass and straightedge as a practical computational device, as witnessed by the efficient epicycle approximation scheme for celestial mechanics computations.
Many conjectures and generalizations of solved conjectures from graph theory should be amenable to solutions by collaboration. However, typical experience with collaborations suggest that teams of 2-4 members are far more effective than significantly larger teams. An example of a very successful team in this area are N. Robertson, P.D. Seymour and R. Thomas, which attacked problems like the strong perfect graph conjecture, generalizations of the four color theorem, and graph minor related conjectures. The elapsed time between announcement of new results and their actual publication has been notoriously long, also for other teams of researchers in the same area, indicating that pure workload volume here is slowing things down, so that collaboration (which already happens) could be beneficial to speed things up. (I'm still curious when some of the announced generalizations of graph minor related results for directed graphs will be published, and whether the people who will finally publish them will be the ones which initially announced the results.)
I currently try to understand the role of completeness of intuitionistic logic in practical applications of computer assisted proof refutation. But if you really plan to do proofs by massively online collaborations, then having a solid computer assisted proof refutation system in place might really be important. After all, if you don't know your collaborators sufficiently well, how will you be able to judge whether you can trust their contributions, without wasting a significant amount of time checking everything they did? (I have the impression that mathematicians are more used to proof refutation and enjoy its positive sides like direct personal feedback, while computer scientists show less routine with this sort of feedback.) Anyway, establishing that proof refutation is a natural part of collaboration when tackling difficult problems seems like a good idea to me.