Measuring a potentiometer with noisy power
What you are looking for is called an "instrumentation amplifier". These amplifiers measure the difference in voltage between two inputs with a high degree of common mode rejection. You can get instrumentation amplifiers in dil packages suitable for use on perfboard easilly enough.

The two wires to the instrumentation amplifier inputs should run very close together to minimimise interference, a twisted pair may be a good choice.

You need to make sure the supply rails to the instrumentation amplifier are wide enough that the two inputs do not stray outside of the amplifiers common mode inputs range.