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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.

Myself I wouldn't be surprised at all to see 54 year old (from 1965) mechanical tachometer with 5000 hours on it starting to break down from all that age and vibration.

You can get a new TSO compliant replacement unit for as little as 250 bucks...

https://www.aircraftspruce.com/catalog/inpages/mech_tachs.php

...so the broken one in the plane is likely headed for the trash bin. You'd be nuts to get it overhauled unless you can get someone to do it for that price or less.

This exact chain of symptoms happened to me on an automotive speedo - wobble, screech, increaing to extreme wobble, needle falling off, odo still working... exacerbated by low temperature in my case. This is due to old lubricant not having the correct viscosity due to hardening up from age. It was like this for many hours. I put a drop of oil in the right place, and it was fully cured and the repair held for over a year so far. The needle snapped back on, but it took some iterative testing to get it to point correctly.

So I expect your unit should be very repairable.

Of course, being aviation, this work must be done by someone certified. The crux of the work would be comparable to what I did, but with proper testing and formalities.

The most likely cause of this problem is a lack of lubrication in the drive shaft for the tachometer. It might be fixed temporarily by disconnecting the tach drive shaft from the tach head, removing the drive shaft, lubing it, and forcing some oil or grease down the shaft.

Or you could replace the entire drive cable assembly.

Please note that this level of maintenance is not allowed to be done as preventative maintenance by 14 CFR Section 43. Legally you will need a sign-off by a repairman holding an Airframe and Powerplant license, aka an A&P.