Having non-profit collecting crowd-funds for the solutions to the important scientific problems would be great. But I would imagine that for those most capable of solving these problems prizes are not essential, while grants that would allow them to conduct the research in the first place are.
So... Create capable non-profit with goal of crowd-funding important research.
The key innovation would be an investor (venture capitalist?) willing to give out grants themselves, in return for a share of the resulting X-prizes. Then funding the prizes would, through Market Magic™, produce grants.
So... Create capable non-profit with goal of crowd-funding important research.