I have had one of my players ask me to run specific games a few times, though those were always games I was already running elsewhere and had told her enough about that she'd gotten interested. Other than that, I'm usually the one to decide I want to run something and then ask around to see who might be interested.
Though one of my current ongoing campaigns came about as follows:
1. I suggested, based on the known tastes of the players, that we play a 1920s pulp game, like Spirit of the Century.
2. All the players agreed and we started discussing character options.
3. One of the players, just to mess with me, asked if he could play a talking rabbit.
4. I, just to mess with him, announced that in that case, we were playing GURPS Bunnies & Burrows.
5. We have now been playing GURPS Bunnies & Burrows for a full year, and it's been one of our more successful campaigns.
No, I'm not sure exactly how we went from 4) to 5), either. It just sort of happened. :p