External influences with ADC
Some things to try:  

Increase the value of the bypass caps on Aref, Vcc, and AVcc (p20) of the 328P.  

Insure that the supply and ground path of the variable resistor are not directly inline with the supply or ground path of the LEDs. 

Reduce the load on the 328P by driving the LED's with a transistor or additional gate. 

For further critical noise reduction the Atmega328/P data book contains additional improvement methods, see the ADC chapter.
http://ww1.microchip.com/downloads/en/DeviceDoc/Atmel-42735-8-bit-AVR-Microcontroller-ATmega328-328P_Datasheet.pdf