I'm a little surprised to report this, but children can strength train more or less just like adults, provided that they receive competent instruction from a trainer knowledgeable about physical development, and that the training proceed slowly. Obviously, it's also of the utmost importance that the children are adequately nourished and hydrated, and that proper safety precautions are taken.
During a growth spurt, or "peak height velocity" (which occurs around age 12 in females and 14 in males), kids experience temporary bone-weakening, muscle imbalances and tendon tightening, and are at increased risk for overuse injuries. This is why it's crucial for children to train under the instruction of someone qualified to assess their biological age, as opposed to age in years and months.
Emotional maturity is also important, and it's a good idea to establish a training program that is noncompetitive in nature. As youth sports become increasingly specialized at younger and younger ages (many kids these days are encouraged to train one sport all year long, making it impossible for curious walk-ons to take up the sport against these Robo-babies), youth strength progams are springing up in well-funded suburban gyms everywhere. Check out www.strongkid.com
for more info.
Now, for my two cents: I say let them run around the yard some more. They have the rest of their lives to get buff.