You could put a Staton-Inc(.com) axle mount motor kit on it no problem. His axle mount kits are just what they sound like and mount the motor to the left side of the rear wheel axle rather then in the frame triangle.
For a kids bike his 25cc 1.1-hp Robin-Subaru kit would probably be the ticket. It's the lowest powered motor he offers and it's a four stroke so no mixing gas and oil but your kid should still be able to go faster then he can pedal.
The combined weight of me and all the gear I haul on my bike on my daily commute is in the 300-lb range and I'm using his axle mount 35cc 1.6-hp kit and it pushes me along just fine on the flat at a little over 20-mph when I don't pedal at all and running both the motor and pedaling in top gear I cruise at about 25-mph.
Se no reason why the Staton 25cc axle mount kit with a 13 or 14 tooth drive sprocket on the gear box wouldn't push your considerably lighter weight son and his bike down the road at similar speeds.
Here is a direct hot link to the axle mount kits I'm talking about:
http://www.staton-inc.com/store/cat..._Gear_Chain_Drive_Engine_Motor_Kits-14-1.html
The speed calculations on his standard list are calculated for a 26" wheel but if you would just simply use his provided note fields to explain that your going to put his kit on a 20" wheel BMX bike and you want the alternate mounting bracket for the smaller wheel size and a 13 or 14 tooth drive sprocket to compensate for the smaller wheel size. His kits are among the best quality and are very reliable which is a major concern for me as a daily commuter.
Don't know if that would work for you or not but I suggest it because his axle mount kits don't require frame space for the motor and have been used on wheel sizes all the way down to 16" and he already has alternate brackets in stock that hold the engine up higher for ground clearance on the smaller wheel sizes and all you have to do is ask in the notes and he sends you the alternate small wheel bracket.
Obvously you could also put a 35cc size 1.6-hp four cycle motor kit on your kids BMX bike like I have on mine if you think he can handle the extra power and since he is (presumably) a lot lighter then me and my gear he could go faster then I do with a bigger drive sprocket. You could go with a more powerful two cycle motor kit as well but the four cycle kits are so nice not having to mix gas and just being able to fill up anywhere.
Another potential nice thing about the staton axle mount kits is that you as the father can adjust how fast he can go under motor power just by changing out the drive sprocket on the gear-box. With a 20" wheel the smallest size 10 tooth sprocket will only let him go about 17-18 mph when the motor is wide open throttle at its highest RPM and if you put a 16 tooth sprocket on that motor RPM regulated top speed jumps up to 27-28 mph with a 20" wheel size and all the sprockets inbetween adjust incrimentally assuming the motor used has the power to spool all the way up to its full RPM pushing the weight of your kid and his bike.