Thanks for the tip on the alternately blinking colored lights on the rear view mirror. I did the following, which worked in my car. I don't know if it will work for any other alarm system.

1. There is a little toggle switch hidden in the left well of the driver's seat. The forward position is for resynch in my car.

Be careful with this switch, because a combination of the switch movement with the driver door open or closed or some other things can change the alarm setting to "valet" and do other unwanted stuff. I don't have a manual for my alarm system, and it took me a hours of googling around and trial and error to figure out how to fix this last time I messed with an alarm and got it upset.

2. Removed negative terminal of battery (I actually already had a knife switch installed on that terminal, so the battery doesn't discharge if I leave the car sit a long time).

3. Wait 10 seconds.

4. Replace negative terminal. Alarm will sound.

5. Within 5 seconds, press lock button (the II botton on a two-button remote).

6. This should resync the remote. In this case, I used a different remote with a better battery for the resync.

7. Then I locked and unlocked the doors with the remote.

8. Then I locked and unlocked the doors with the regular key.

9. When I turned the key in the ignition (driver door closed), the rear view mirror lights didn't flash, as they had before. So I turned the key to start, and it fired right up.

10. I put the remote into the glove box. From now on I intend to use the regular key to lock and unlock, and forget the remote and the alarm. I have a steering wheel lock bar that I will use if I take the car anywhere it might get stolen.

11. I called the local Mercedes dealer, who says that they can't remove the alarm, but they can disarm it is such a way that the remote will still control the door locks, but that everything affecting the engine starting will be disarmed permanently. They estimate about $200 for the job. I figure it will be worth it not to worry about the alarm acting up and stranding me someplace if my resynch trick doesn't work.

In the above I modified several variables at once, including the remote and using both the remote and the key to lock/unlock the doors, so I don't know which of all the things I did actually cured the problem. But from now on, no alarm for me, unless I replace the current one with a new one where I can actually have a manual and technical support.