Thanks for all this good information. I also have a Gentex 261 mirror from a 2003 Avalanche. This mirror is fully loaded with Onstar, auto dim, Airbag display, compass and temp. In my case no temp, compass, auto dim and airbag display. It would intermittently come on and turn off on its own. The power button did nothing, but the Onstar buttons did work.
This forum inspired me to take my mirror apart. I also bought a used Gentex 261 mirror with the cable so I can power up my broken mirror to see what was going on. The guys who mentioned that the mirror would work if they raised the voltage gave me the idea that there could be power issues. Most electronic integrated circuits work off +5 Volts. I searched around the main board and found the regulator that takes in the 12 volts coming in and cuts it to +5v. My +5 volt regulator was putting out only +4v. I decided to change this part so I ordered ten 78M05 regulators on Ebay for $4. Ten was the smallest quantity you can buy of this part at the time. 78M05 is the part number of the regulator. I replaced the regulator and now I get the required +5v for the rest of the board and everything works like new airbag, auto dim, comp and temp.
The regulator is on the main circuit board near the comp/temp display board. Sorry I'm new to this forum and this site won't let me upload pictures of the board and 5v chip. Anyway if you don't know what it looks like this regulator is black and has 2 pins at the bottom and a small heat sink at the top. It's about 1/4" square. You should see 78M05 printed on this part. The pin to the left should measure around +12.5v and the output pin to the right should measure +5v. If the output pin should measure much less then +5v change it.
This is a surface mount chip so be very careful to not interrupt the other parts around it. I have a hot air rework station to remove this part which make it very easy to do and you can remove this chip in one piece as apposed to cutting it, but you can also use a soldering iron or hot air gun like the one you use to remove old paint if you have that. Just be very careful not the melt anything around that regulator.
Hope this helps others with their mirrors.