Detailed question. Deserves an answer.
A wire dimensioning/diameter problem usually can be easily solved using a wire gauge calculator (just google it there are plenty).
Usually the longer the wiring the thicker it should be - even though there will not be high current as it's just a poti (variable resistor) to an ADC for the \[\mu\]Controller. Unfortunately this means the pin voltage (thus the poti wire voltage level too) will not be higher than 5V, if not even only 3V. Another reason to better use too thick a wire than too small.
The wire size calculator should give you a good idea. Nevertheless I'm not sure about noise immunity - after all the analogue signal is going to be converted to a digital value (by the ADC hardware), and you don't know if there is any filtering or if it is enabled in software or not. So a cap for smoothing the signal could be necessary (signal to ground). Anyone correct me if I'm wrong?