The Mars Society has started a Mars Project Challenge, which is an effort for Mars Society members to suggest ideas for the next project the Society should do to "take it to the next level". It seems like a pretty ambitious level as space society projects go, too, if they have the funding to match their proposal. There isn't a cash prize for the winner, if one is chosen from the suggestions, but the satisfaction of knowing your idea is being used.
Since this is Space Prizes I'll note that there are a lot of ideas already out there in the prize world that might work for the Mars Society. Their own University Rover Challenge could form the basis of a more ambitious Mars rover prize. The idea of the Mars Robotic Construction Challenge held at a Mars Barn is a similar idea, and could get even better if some Centennial Challenges money could be added. Another possibility would be for the Mars Society to pitch in some bonus prizes to give some of those existing Moon-oriented prizes like the Lunar Lander Challenge, Google Lunar X PRIZE, Regolith Excavation Challenge, or MoonROx Challenge a Mars twist.
Of course there are also a lot of non-prize ideas that might also work. Maybe they'll come up with something they can actually get into suborbital or even orbital space using commercial space services, or even hitch a ride to Mars on a government mission.