Hmm. I see an opportunity here for something that combines taxpayer pay-in with crowdfunding-style pay-out. Imagine if the NIH set some percentage of their budget aside for matching "donations" to crowdfunded science projects, at an extreme ratio, like 100:1 -- so $30k of demonstrated public interest would be matched with $3MM of NIH funding.

That seems, to me, to be within the spirit of representative democracy: for every 1 person who knows enough about X to put their money on the line, there's probably 100 people who would if they had X explained to them.