Update - I got a better camera and recreated the experiment
I recently acquired a garage parking space with a remote opener. It's been about ten years since I've had such a luxury, so you can imagine my (slight) disappointment when I found out what horrible range my remote had. I had heard stories of people extending the internal antenna, but that's no fun, and it increases the range that your opener will receive a signal from, which is never good. I decided that the same antenna extension technique must be applicable to the remote just as much as it is to the actual device.
What I found was actually much simpler than I expected, and depending on your remote type, this might just work for you too. Once the unit was open, it had the usual bunch of diodes and such, but there was one strip of copper that didn't seem to serve a function, so I thought logically is must be the antenna, but I wasn't about to get started until I was sure.
Every device in America that transmits a signal must be approved by the FCC and issued a FCC ID. Using this FCC ID
I tracked down two models of my opener. One was manufactured in 1989 (which is mine) and the other was made in 2001. Unfortunately the schematics for the 1989 version were unavailable, but the 2001 version was fully documented, and to my delight it was the exact same circuit board manufactured by a different company. I'm sure the software in the chip may have changed, but that antenna was clearly labeled, and just where I originally guessed it was. Off to work!
For my antenna extender I thought I should be safe and get a hold of material that had already been used as an antenna, and was made of copper just like the antenna on the circuit board. So I cut off a 4" piece of flexible radio antenna from an unused stereo system, stripped the plastic coating, and I had my extension.
The first try at soldering proved unsuccessful. I won't go into detail, but at one point a bead of liquid metal was cooling on my thumb. For the second round I decided to play it safe and get glasses and a clamp (something I should have started with).
The second try was a charm and I had my extender. I proceeded to re-assemble the unit, but first I decided to carefully coil the extension around a plastic support column on the outer chassis. This should increase range, and provide a secure spot away from the other circuitry.
Before I had started this project I tried testing the remote from two walls thickness with no result. After it was completed, the same test resulted in success. I have yet to test the extent of the remote's range, but I am definitely pleased with the results.