This is because of noise and ripple.
Place decoupling capacitors across all VDD and VSS pins. Place the capacitors as close to the VDD and VSS pins as possible.
Between the pot wiper and the AN0, connect a 1k resistor. Connect a 1nF-100nF capacitor from AN0 to ground.
At the input and output of your +5V regulator, place decoupling capacitors (10nF - 100nF) in parallel with electrolytic capacitors (10uF - 100uF).
At the output of the regulator, place a bulk capacitor (220uF - 1000uF).
Also, at the input, you may place an inductor in series (between the power supply output and the regulator input). Connect a capacitor (10uF - 220uF) from the regulator input to ground.
Hope this helps.
Tahmid.