I've played some games I was familiar with, as well as a few new ones, including Yinsh, Cavemen: The Search for Fire and Kings of Israel.
Some were better than others, but the most influential so far has been the Pathfinder card game.
The idea of a co-op game where you can not only customize your deck to match your own style as play but help your teammates during the post scenario trading phase seemed different to me, and I started thinking about implementing and expanding this idea into video games.
I guess MOBAs have some of these elements of helping other people build their characters and form strategies to defend and attack on multiple fronts, but I think it could be pushed even further - if killing mobs gave out experience points to the entire team, which needs to decide on how to distribute it.
What if different characters were suited for different tasks, which would unlock equipment best used by team members going after different goals?
How about players unlocking abilities for their teammates but not themselves?
A game based on combat and magic linking all these abilities seems like the easiest and obvious choice, but I've also given some thoughts to some more unusual themes such as city building, or flying an airship.