It's probably the ring resistor, which is underneath the shutter speed dial on the FM and FM2/2n cameras. Would be easiest to find a camera with a bad shutter, but a good meter, then swap the shutter speed control assembly, complete with the ring resistor to your camera. The service manual can be found online, so, since you plan to do it yourself, you're only out the parts and your time to do the job.
Before doing that, though, try adjusting the ASA to something like 1600 or 3200 and see if the meter responds. It's possible that it's just WAY out of adjustment. Looked at an FM recently that had this problem. The red LED next to the "-" was lit constantly, until I adjusted the film speed up to 3200. Then, it acted as if it had been adjusted to 100.
These days, the FM is like the Nikkormat FT series. Cheaper to buy a good working example than to have a shop fix one. So, I don't buy them if there are any meter issues, or if 1 second seems slow, etc. The FM2 and FM2n are getting borderline for bodies that aren't at least KEH EX or EX+ grade. When you can buy a decent black FM2n in good working order for $120-$150, it doesn't really make sense to get a less clean body overhauled, unless it has sentimental value.
-J
