I have a three year old. At two he liked to play with the little $20 Walmart toy laptop. He also liked to stand on it and otherwise abuse it. He also liked to play with musical instruments, and any number of other things he saw me do, because that's what kids do.
Yes, he's a mimic. That's what kids do. That's how they learn. He sees me on the computer; he then wants to be on a computer. He sees me play guitar, he wants to play guitar. What he sees me do he gets interested in, then I help him and show him how he can do the things he sees me do.
It's called parenting.
As a result, he can now at three and a half quite easily navigate the noggin website and the pbskids website. He can play the games, he can change games himself when he wants to, and when he gets bored he says he wants to do something else and then we do that. We limit the time, because he doesn't need to be on the computer all the time, but he self-limits too. By his fourth birthday he'll have his own system in his room, set up with a couple of kid friendly programs and web links. He'll also have his own bike and know how to ride it, and will hopefully have his letters down enough to read a bit, and speak a little more of a second language than he already does (since we're starting to learn Chinese together). I don't think he's a prodigy, but I'm also quite convinced that any kid can learn faster than what some of the posters here seem to think a kid can "handle". Most kids are brighter than most adults give them credit for.
My advice for the op is get an old system and set it up with a kid-oriented *nix distro, and help the kid understand what to do with it. My advice to the posters saying the op should get a life is that it looks like he has one and is trying to give his kid one too, so maybe you go a little heavier on the constructive and a little lighter on the criticism.