Daughter, Speaking in first-person terms, my father's successor's father would be. Then, I am my father's son. So my successor's father is my father's son, which relates back to me.
Being that I don't have a son or a brother, the successor to me would be my daughter. This is a riddle, which would involve advanced thinking. It can also be found on tests to find one's IQ. Some may find it confusing at first, but as it is broken down, the answer is easily found.