Op says "not smooth" Gator says "material is odd" (agree). Also agree with Gator that location found is a great clue. Buddy says asphalt cores get distorted from heat...could explain off center small hole..
never seen a discodial or game stone that's rough, has a hole and made with that material but I would not rule that out. Association would be a great clue. Fwiw...I have witnessed very few discoidals/game stones, my mind is very open on this piece.