I have some discussions of running for my son (started when he was 4; he's 5 now) at
http://www.enworld.org/forum/story-...s-my-sons-first-game-updated-7-18-7-31-a.html
A couple of general thoughts:
Use a simple system for young kids. 4E was not a good choice for a 4 year old.
For young kids, physical representations make the game work better. Minis, pictures, etc., make it easier for them to picture what's going on. As they get older, that's less necessary.
For young kids, short games are the rule, not the exception. My games for my son are typically an hour or so long--maybe 3, 4 encounters. Both keeping the overall amount of time short and keeping the number of separate things small are important--you don't want the child getting bored or lost. With a seven year old, it may be less of a concern, but still--don't start with epic things.
One-on-one works well for young kids. I've tried group games as well, and those were fun, but they were also way crazier. It can be hard enough to keep one kid focused--two kids at the table makes lots of opportunity for somebody to be bored, spaced out, or disruptive. Still, once you get the one-on-one working, inviting one of your kid's friends could be a good idea.
And my biggest conclusion: you can often run a good game for a young child by asking the child what they want to do, and then tailoring the adventure to that. Because the adventures don't need to be terribly involved or complicated, you can fill the details in on the fly, tailoring them to what your child suggested and then adding in new elements.
Have fun!