I would definitely go for "online games". "free online games" is also recommended and so does "online multiplayer games". Just run search at google keyword tools and you will see what I mean here. Make a good choice and stick to it.
No problem. I hope that can help you and your site to make it popular. Just stick to it and monitor the results using analytics. Analyze and study the traffic that each keyword brings in and if that does not work, you might want to try using other keywords. Good luck.