Wired is always better if you can run a cable but wi fi should work fine for an internet connection, it just wont transfer from pc to pc as fast as a wired connection can.
If the wi fi connection is flaky its because theres something interfering between the wi fi on the router and the wi fi enabled client, maybe the construction of the walls/floor.
If you cant get another cable from the router to your sons room then you may be able to get a wi fi extender placed somewhere that will improve the wi fi capability to his room.
Have you tried one of your wi fi connected laptops in his room to see if the connection is stable on that ?