I've had this happen before. It was one of my old ThrustMaster or CH sticks that did it. It was just a spring that popped off so I opened up the stick, reattached it, and it was fine.
I'm not sure about the insides of the MS Precision II, but I know it uses pots unlike the 3D Pro and Precision Pro which use digital/optical sensors.
It's broken anyway. I would just open it up and see what can be done with it. You can't really make anything worse.
16 years is a really good service life for a stick that uses pots. You could always try visiting GoodWill or a thrift store to find another. I see used sticks for sale all the time. I've found 3D Pros, Precision Pros, Precision IIs, Saitek, CH Products, ThrustMaster, SunCom, Gravis, and all sorts of old input devices.
About a month ago, I found one called a "InterAct Raider FX". I had never seen one of these before, but it was in really great condition in its original box, so I grabbed it for $2. It's a digital USB stick that has some rudimentary force-feedback/rumble mechanism. I researched it a bit and found that it will only work with Win9X. I tested it out and it works fine; no spiking or slop in the center position. I haven't played with it too much yet, but it actually seems to be more accurate than a 3D Pro or Precision Pro. It's too bad that its drivers won't work with XP, but I'll enjoy it for what it is on my legacy systems.