Got Fios installed with router on main computer. Son's computer is in basement. Do I need a separate router for him to be able to go on the Internet?
NO. But you either have to have a wired connection run from the router or perhaps wireless connection. Other possibilities exist with additional equipement.
the internet signals are also carried on the coax. If you have coax in that room for TV (and you ordered TV), then you can use a splitter and MoCA adapter to get Ethernet into that room.