When you say online games, are those web-based (http/https)? Or are they in-app/game connection? If the games are web-based, you'll find it hard to block the port numbers (80/443) as these ports are used by legitimate connections. It'll be pretty hard to block the domain names of those games, as you will have to get the list of all online games. Or you can block all allow some by default. That is, block all websites and allow only the legitimate ones.
If your router supports regex (regular expressions), you can block certain keywords. Depending on the capabilities of your router, you can also use authentication for users. My suggested solution? Get a low power cpu and implement a proxy/firewall using BSD or Linux (there are easy solutions like pfsense or endian fw).
