I got the bug to tinker last night, so here's what I came up with.
I pulled the vanity light assembly out of my driver-side visor, gutted it, and tucked a very small garage door remote into the void. The opener is held in place with a couple of smears of RTV...
I then modified the mirror mounting so it could slide a bit, but is held in place with an elastic band. Now that the proof of concept works, I'll replace with dabs of silicon.
Now the vanity mirror looks just like a vanity mirror, but if you press the center, the garage door goes up and down.
http://www.youtube.com/watch?v=L5UocnPvc6E