TORONTO—Ever since the Wright brothers took flight in 1903, feats of aviation have come fast and furious—from Charles Lindbergh's trans-Atlantic flight in 1927 to the first, round-the-world, nonstop balloon voyage in 1999.

But there is at least one big milestone still out there: In 1980, the American Helicopter Society International created a $25,000 prize for the first aviator to stay airborne for 60 seconds, reach an altitude of more than 10 feet, and hover without drifting more than 1,076 square feet—all in a machine powered...