How to measure servo potentiometer
What is connected to the other two leads of the pot? 

The linked page is vague about how that's handled. If it's power and ground, how clean is the power in question - if it's the same power that's being supplied to the motor, it may be a bit filthy when the motor is running, anyway. If you were "following" that vague page and the center pin is the only thing you connected - well, there's your problem - give it clean power and ground and things will get much better.

Try some capacitance (aka, a low pass filter) between the ends of the pot on the one hand, and between your signal input pin and ground on the other. 

If you have access to an oscillosope, look at the signal from the wiper (center pin), and at the supply (across the end pins of the pot.)