OpenNASA invites you to contribute ideas on making NASA more efficient, innovative, and open within the agency and to the outside world. You can discuss and vote on other contributors' ideas, too. Some of the ideas there feature prizes or competitions of one sort or another. Here are a few examples:
Create a Science Data API Challenge
Junior Astronaut Corps - Allow students to compete for 25 coveted slots in the Junior Astronaut Corps -- using much of the same criteria (but age-appropriate) as we use to select astronauts. ...
Millennial Challenges ... Centennial Challenges on Steroids
Step-by-step prizes for Flexible Path goals
Incentive awards 4 creative thinking (new participatory office)