Well, it's done and it works! The guts of my garage door remote now resides in my windshield surround, wired into the mirror circuit, activated by a small Radio Shack switch!
Here are the tools and supplies I used:

Soldering iron, solder, wire, remote for door, wire stripper, screwdriver, and momentary switch from Radio Shack ($2.49).
I first disassembled the remote, removing the clip, then the screw holding the case together.
Here is the back of the circuit board with the locations (on mine) where the wires are soldered. The two to the switch are where the original switch on the board is soldered. Check these with a multimeter and see that the voltage goes from 12 V to essentially 0 when the switch is depressed. The two locations at the bottom are simply the battery terminals. Be sure to note which is "+" and which is "-".
This next picture shows the board taped inside the winshield surround trim piece. To the lower right of the circuit board is the Radio Shack switch which is placed through a hole made in the trim (Dremel rocks!). The switch is soldered to the (now) underside of the circuit board, and leads are soldered to the two battery terminals.
Here is a picture of the wires to the mirror, a small portion of the fabric tape that covers them split and pulled back, and two Posi-Taps installed on the blue w/white stripe (12 volt power) and black (ground) wires. (A BIG thanks to AztekzRPurty for putting me onto posi-taps!

). The red (power) and black (ground) leads from the circuit board are then fastened to these taps.
Finally, here is a pic of the finished install. The button is small and inobtrusive (note my tweety switch by the other visor!). The remote is powered from the car and so I never need worry about replacing the battery (or losing the remote, for that matter.

)
IMPORTANT NOTE: THIS MOD WORKED AS SHOWN BECAUSE MY REMOTE IS 12V. Yours may 9V (or 3V or 1.5V), and you will need additional circuitry beyond my simple install.
I hope you have enjoyed our time together. There will be a quiz next week.

NOT!
--Chemist