First of all, what do you mean by "mopeds?" Actual pedal mopeds, or small scooters? Do you need a certain speed to keep up with traffic? If you have steep hills, I don't recommend a small scooter, their CVT transmissions do not like hills. A 50cc 2 speed Tomos moped will easily outclimb my Vino 125, if you back off the throttle and let it stay in first gear. A small motorcycle, say 125cc, with a manual transmission will blow by a 125cc scooter like it was standing still going up a hill. The MadAss 125 should work if it is a manual transmission. If you seriously want something your kid can use as a dirt bike, then a small displacement dual sport is the way to go, just make sure the kid stays off the road with it. No 11 year old belongs on the road, though I was riding a dirt bike on rural country roads at that age.
I won't spend more on a bike than I think it's worth, but if it's a good deal, I don't seem to have a problem buying bikes I don't need.
2002 Vulcan 750, 2013 Royal Enfield B5
2001 XT225, 2009 Genuine Stella
1980 Puch moped