Why would a C150 tachometer bounce so much that the needle fell off? On a long flight in a C150E, the tachometer started bouncing a little bit and the severity of the bouncing gradually increased until it was extreme: The tachometer was spinning in loops and making a loud screeching sound. 
 The counter continues to increment.

Here's a video of the tachometer in the last 10min of our flight. Just after that video, the needle fell off, but the screeching sound persists and the metal pin at the center still seems to be spinning:



What could cause this?

I understand the tach needle is not powered by any gear mechanism, but rather a small alternator that converts the angular velocity of the tachometer cable into a current which exerts a force on the needle.  This should be highly reliable and I don't see how it could cause this behavior.
Cessna 150s have a mechanical tachometer driven by a mechanical tach drive cable geared to the engine. 

The tach needle is moved by sensing spinning magnets driven by the tach cable.

They do wear out over time and yours has reached the end of its life.