Parents usually ask for cranks that are much too long for their children. They worry about their kid outgrowing their cranks, when they shouldn't.  For pre-teens I recommend 10% of the child's height. This does allow some growing room as adults do best with cranks about 9.5% of their height or less.  If it's near the end of the riding season where you are I'll go another 5mm if the child will be off the bike and growing for several months.

Because we are used to thinking of cranks only within the range of 165mm to 175mm, 150mm sounds very short.  Therefore, many people think 150mm would be suitable for a typical 7 or 8 year old child.

  This is a photo of 175 and 135mm cranks.  Look at them and then stand in front of a mirror with your child.  Maybe cranks in the 110-140mm range don't seem quite as short now.

Plenty of testing has shown that everyone has a critical length beyond which power drops off drastically.  This is because a muscle's ability to exert tension falls off when they are stretched beyond about 120% of their resting length. If your child's cranks are too long they will be inneffective through much of the top part of the pedal circle. 

OTOH a lot of testing and real world examples show that you can go considerably shorter than this length with no loss of power. This is because your knees are straighter so you can push harder on the pedals. And because your muscles are in the length range where they can apply peak tension through

So, as a child grows to a height where they COULD ride longer cranks, they aren't giving up anything by riding the cranks that were fine for them a year or two ago. On the other hand a child riding cranks sized for the height he or she will be in a couple of years will not only be slower but also risks knee injury. About 15% of my customers have damaged their knees riding cranks that were too long.  Possibly because the muscles that stabilize the knee have also been stretched too far to apply sufficient tension. Also, smaller tandem stokers have a hard time spinning the same RPM as the captain if their cranks are too long.

I had a customer who had tried for years to get his 5’ 1" wife to try short cranks. But she had been told she needed “leverage” for the mountainous terrain where they lived. I sold him 2 sets of 120mm S600s, for their 8 year old twins. The 1st ride on the new cranks, the girls were suddenly out-climbing mom. The next day she had him install the 152s that had been on the girl’s bikes on hers.