I used to be in an FC that would have mount race events. We would have it in the housing areas, and have a 'race track' set up so we knew the route, then limit it to four or five laps. The mounts all had to be similar, unless everyone had the motorcycle, then everyone could use it (the inherent speed buff built into that mount) but otherwise, it was normally chocobo racing. You wouldn't think that it would be competitive, but it was actually a lot of fun and the outcome was always different every time we ran it. The reward was picked by the FC leader and was normally a housing item that was hard to get (use the maps to collect some rare things to use as rewards) or rare materials.

You could also put your workshop to use and have a competition to see who can donate the most common materials for a project. For the reward, a winner could be awarded the project like the housing project items for their house, or reward them a mount or something if they don't have one yet. Rewards could also be outfits that are crafted, or bought or won from Firmament.