I hate games that put you under time pressure, so I'm trying to think of a game that has this "one minute" theme without it affecting the gameplay as a timer. I don't want a countdown or a count-up. 

At the very least, I think I will make any time-based mechanic a bonus rather than part of the core gameplay.

Anyone else thinking this way?

I guess I could always look as "one minute" as a measurement in degrees rather than a measurement in time!

Well, I also think, that timer is just to basic. Why not to make the game get harder each minute? Or you change weapon type each minute? Or, it's 1 hour 1 minute... Think about it ;)