A clean-tech start-up that thinks it has solved a problem that has vexed engineers for decades--how to effectively store energy generated by wind and sunshine--has gained some heavyweight financial backers.

An investment group that includes Bill Gates and Peter Thiel, the co-founder of PayPal and the first outside investor in Facebook Inc., has poured $37.3 million into LightSail Energy, a three-year-old start-up dreamed up by a Princeton graduate school dropout. Thiel led the investment.

Danielle...