Assurance contracts? http://en.wikipedia.org/wiki/Assurance_contracts
Example: you start a project on something like kickstarter, people contribute the amount of money they like. Then after the project reaches its goal, the research starts. If goal isn't reached then people get their money back.
That probably improves things - certainly, it eliminates the risk of "I gave and the research didn't happen anyway" - but (as typically employed) it does not eliminate the issue: the odds that my contribution will be deciding is low.
If you had a contract that required defined (possibly in terms of wealth/income) contribution from everyone and would only go if unanimously accepted, that would eliminate this problem but raises obvious logistical issues.
Let's imagine a hypothetical piece of research that every single person wants, and wants enough to willingly contribute their share (per capita, or wealth/income adjusted, whatever) if that is required in order to have the research done. However, if they value the money they would be contributing more than the marginal change in the research for lack of their funding, they might rationally choose still not to contribute unless coerced to do their part.