I started on a Honda Odyssey when I was 7. Got my first bike when I was 9 (a real lemon - Yamaha MX80, 2-stroke with oil injection), then an XR100 when I was 11. Stopped for a while and then bought an XR250R when I was 22. Don't have it anymore, but do miss it on occasion.
I've seen a number of options for the fledgling dirt rider:
The ever popular Z50 - not much of a dirt bike though.
There's those little 50cc automatics, not sure of the designation.
There used to be a Honda CR60, maybe a little much for a 6 year old, though.
And, finally, when I had my XR100, there was this other kid who had a YZ50. This thing was a rocket. We were neck and neck in a straight line drag race. Definitely over-powered for the size of the operator that it was designed to carry.
One thing I learned early, was how to "abort mission." I don't think launching yourself clear of a 4-wheeler is as easy as it is on a bike.