The most effective big-bore engine is the 250cc four-stroke. By going to a bigger piston, enough to get it up to 262cc, you get a two horsepower gain. On the second place comes the Kawasaki KSR110, with a 4mm bore up.
For some four-stroke engines, which are in basic already a bore up when the leave the factory, the engine's tipping point is much lower. For example the Honda CRF250's tipping point is 256cc. That is a 1mm increase in bore size.
For the bore up of Kawasaki's popular KSR-110 and other smaller play bikes, most of them 150cc and smaller play bikes use cylinders with steel liners. Even with the benefits of a steel sleeve, it's not recommended to overbore more than 2mm. However, if you're already having the cylinder bored, it's just as easy to punch out the old sleeve and replace it with one that accepts up to a 6mm larger piston.
On the popular Kawasaki KSR110, a 4 millimeter increase in bore results in a 143cc displacement. What do you get for this modification is a good 1.5 horsepower more at the rear wheel.
For the Kawasaki KSR110 you have to change the jetting. The smaller the engine size, the more jetting help it will need. With the KSR110's you need a carburetor upgrade to go along with the big bore. If you really want to make the most of that 143cc kit on your Kawasaki KSR110, you'll need to switch out the stock 18mm Keihin for a 26mm Mikuni or 28mm Keihin.
On the most other motorcycles, the stock carburetor will almost always support the 2mm bore increase. So much so that in many cases you will need to jet down! It may seem counter intuitive, but a larger engine draws more air, which in turn drops more pressure, sucks harder and pulls more gas out of the float bowl.