start with the game type , IE your idea or "vision" for what kind of game you want to make.

from that, a list of game features required should naturally come to you. - as in:

"well... it'll need this, and that, and the other thing - and the player will have to be able to do X,Y,Z, and W.".

then for those features, you come up with a game mechanic where your have the design restriction of "just one finger".

you can make the game mechanics as fancy as you want (IE realtime vs turn based combat) along as you can come up with a workable game mechanic for "just one finger".

lemme guess - touch screen app - right?

for some sort of at least semi-real time combat, a combo of swipes and taps might work.

surely there's been some research done on touch screen interfaces and what works well and what doesn't. white papers on that topic might point you to some pitfalls to avoid, and some mechanics to favor.