The breakin procedure is another of those controvercial issues. But don't let the details worry you too much, as you are dealing with a mass production engine, and not a highly precise machined part.
When you look into the cylinder, there should be cross-hatching marks. Overlapping swirls that are unmistakable.
These little lines need to be there. And when we rebuild, we need to make sure that all the contamination and crud are out of the little nooks and crannys that are on the walls and in the swirls.
A ball hone is used often to clean up the walls of a cylinder. But, you can also use a green Scotch-Brite pad and some elbow grease.
What you are wanting is the cylinder wall cross-hatching to be consistant from top to bottom.
After cleaning vigourously with the pad or a ball hone, take a clean rag with some pre-mix on it and wipe the walls down. You want to repeat the cleaning procedure until the rag comes back completely clean (no black carbon) after wiping.
I personally never use a ball hone. A ball hone will NOT hurt the cylinder walls in any fashion. They are way too tough for the ball hone to do damage. I simply have some reasons that I only use an abrasive pad like Scotch-Brite. I won't go into why I don't do this here, as it shouldn't matter in your situation one bit.
It is possible for a mass production motor to be way off. In fact, the cyliner can be slightly egg shaped. And if this is the case, you will see evidence of an inconsistant pattern on the piston and walls. If you see a scar or some sort of rub marks on the walls, you may have a deeper issue. You may need remachining.
Im my experiences, an 80cc two cycle that is as old as yours will have some major problems, as kids tend to ride the hides off of them. Be certain to get a Yamaha manual and take measurments of all the parts as you go. Find something that is out of tolerence according to the manual, and you need to replace it before going further...or else try to get lucky...and maybe Jr will end up with a new scoot. :)