Assuming the lamp is 220vac and you want to control the intensity then as such you can simply use a traic circuitry, if you want to use a battery then you should use a bulb that is DC drivable so that you do not need a inverter to convert the AC to DC and vice versa.
Also you do not need to definitely upgrade to a ATmega2560 arduino that has over 200kb of flash rather than you can also go for ATmega1284/1280 based designs! thats half the falsh memory soit means saving money if not much but yes saving that god damn hard earned bucks! or maybe go for ATmega644 etc.
Talk with CrossRoads a respected forum senior member you should try his boards ,they are dependable and comes in green and yellow colors!
well most probably(without seeing your code etc!) the 10k pot is used to set the intensity of the motor driver driving the motors using PWM thats analogWrite(); so when you open the 10k pot more then the speed gets more and when you close it the speed gets lowered.