Combine BBQ rub and brown sugar in a small bowl and set aside. Coat the pork with the vegetable oil and then rub dry mixture on all sides. Cover with aluminum foil and place in the refrigerator for 8 to 12 hours. Remove pork from refrigerator at least one hour before cooking.
Soak wood chips in water for an hour.
Set up grill for indirect cooking. Remove grate and place 3 cups of wood chips in an aluminum foil pan directly over burner. Add one inch of water to drip pan and place in center of grill. Light burner directly below wood chips and set to high. Once you have smoke, turn burner to low. Maintain a temperature between 225 and 250 degrees.
Place pork shoulder fat side up on grate over drip pan and close grill. About every two hours rotate the drip pan 180 degrees so the pork will cook evenly.
After 2 to 2 1/2 hours, add 3 more cups of wood chips to aluminum foil pan.
After 6 hours, start checking the temperature of the pork shoulder with an instant-read thermometer. Insert in thickest part, making sure not to touch the bone with thermometer. Once the pork shoulder has an internal temperature of 195 degrees, remove from grill. This could take 8 to 10 hours. Don't worry about how long you cook the pork. Just make sure the internal temperature is 195 degrees.
Cover with foil and let pork rest for at least one hour.
Shred pork with a fork and tongs. Add some of the liquid from the drip pan to the shredded pork.