Typically, I'd say most kids should start actual weight training around 13 to 14 years old. However, this is case dependent. Some kids mature faster both mentally and physically than others and can start somewhat earlier.
For kids 10 to 13 years old, this is a good time for them to focus on fundamental movement skills, motor skills and learning the movement patterns that will be used in weight training. Using soft resistance like medicine balls and bands is an excellent way to begin strengthening these movements.
If a child has gone through this process of learning fundamental movement skills and has been introduced to basic speed and power training, they will transition much better to introductory strength training phases. I've found that there has to be a specific maturity and developmental capacity for a child at this age to understand instruction and get the most out of training.
The key to strength training at this level is developing and refining the movement patterns. These children are not yet lifting like adults. They should be doing high repetitions with lightweight focusing on technique and grooving the basic lifting patterns. Most changes that happen to children at this level are not going to be morphological. But strength gains will come from within their nervous system.
With developing young athletes, one phase depends on the exposure to the previous. This is why I tell over-rambunctious parents to, "be patient." There's not only the progressive training journey but the physical maturation of the athlete as well.
The goal is to match up proper training with maturation levels. And similar to developing specific motor skills at an early age (running, skipping and jumping) the adolescent athlete benefits from early exposure to strength training. This helps them transition into the demands of high school and elite level sports.
A child who has a great training base will be more apt to excel in the weight room through their high school and college years. It also helps with decreasing chances of injury while elevating their sport-specific skills above their competition.