What is the most affordable way to send a 9-year-old child to visit a relative in another city?

Uaually by plane actually. Trains are usually more expensive and buses aren't safe enough. Book the flight well in advance and be flexible about the date and time. This will help get the cheapest ticket. Also let the airline know when you purchse the ticket that it is for a child and they will help make sure he is safe, doesn't miss any connecting flights and finds his way around the airport okay. And get him an ID at the DMV airlines like photo IDs and it will make boarding much much easier for everyone.