Prepare a grill for indirect heat. If using a charcoal grill, build the hot coals on one side only. If using a gas grill, heat one side only to medium-high heat.
Sprinkle the chicken on both sides with salt and pepper and drizzle with the oil. Place the chicken skin-side up over direct heat and close the grill. Cook for 2 minutes. Flip the chicken and continue to cook with the lid down for 1 additional minute.
Meanwhile, whisk together the barbecue sauce and coffee in a small saucepan.
Flip the chicken again so it's skin-side up, move over to indirect heat and brush with some of the redeye barbecue sauce. Close the grill and cook, brushing with the redeye barbecue sauce every 10 minutes, until instant-read thermometer registers 160 degrees F, about 25 minutes. Remove the chicken from the grill and rest at least 5 minutes before serving. Bring the leftover redeye barbecue sauce to a simmer over direct heat for several minutes. Serve the chicken with the redeye barbecue sauce on the side or poured over.
(Alternatively, you can bake the chicken skin-side up on a wire rack-lined rimmed baking sheet in a 450 degrees F oven for 25 minutes. You do not need to flip the chicken.)