Has anyone else experienced this noise in their Prius?

Sure have. It's not a matter of lubricating the linkage though.

The linkage is 'hunting' back and forth because of flaky signal from a position sensor that tells the controller where it is. So the controller keeps thinking it is off target position, and trying to correct. This thread explains it;

No, that wasn't a mouse in the heater | PriusChat

The gen 1 used potentiometer position sensors for a lot of purposes, not just the HVAC dampers, and they all produced famous gen 1 symptoms when they produced flaky signals. When it happened in the accelerator pedal sensor, you got Big Hand Syndrome (like a Big Hand reached down from the sky and slowed the car down). When it happened in the steering torque sensor, you got the Steering Gear Jitters. And when it happened in the HVAC servos, you got this.

Most of those sensors were changed to non-potentiometer types by gen 2, though the HVAC servos didn't get changed until gen 3. So the symptoms described above can also happen in gen 2.

The different style of servo they went to in gen 3 are not without their own problems, but they're different problems.

 

Yup, eventually the jittering back and forth broke the motor shaft during a cold snap, so no heat in the cab w/o reaching down and flipping the linkage by hand while the fan was on. The jittering went away after I replaced the motor.

 

There are three servo motors: one that selects the air source (fresh/recirc), one that selects the air outlets, and one controlling the temperature blend. Whichever one happens to be jittering can usually have the potentiometer traces and contacts cleaned, but if it's too far gone for that, it can be replaced.

 

This is the part I ended up replacing:

01-03 Toyota Prius A/C Heater Temp Control Damper Motor 063700-7440

 

Replacing that one of the three will solve the problem whenever that one of the three is the problem.