It's a noble idea, but doing big science is really expensive (take the LHC or the Manhattan project for example). Stuff like software development of mathematical research is nice because you have pretty much 0 material cost, but it is really the exception and not the rule.

How would you go about doing brain cancer research without conducting the many and necessarily costly experiments (not to mention the oversight and legal support required)?

Frankly, I've yet to hear a credible answer to these issues. It seems that the best you can really do is try to raise awareness (nebulous and often ineffective as that is) and try to support existing research teams. Or if you are extremely wealthy, you can just fund your own lab to work on whatever you think is important...