I suggest checking out Roblox (http://www.roblox.com/) if your child likes Minecraft and is in elementary school. It is more intended for younger kids (ie elementary age kids) the site is a bit more monitored. Since I've never played either I can't tell you the differences. Roblox is similar it's an online game where you are building worlds with virtual blocks. Both these games are online games where you play with whoever else is on at the time.

My son used to get frustrated in Minecraft because other kids would tear down his creations. My understanding is you can play for free, but if you want your own private space you have to buy some sort of subscription. At 15 DS has moved on and isn't interested anymore is this type of computer game anymore. He mostly fantasy/adventure style games.