The NSCA's (National Strength and Conditioning Association's) position statement for youth resistance training (kids lifting weights) states:
"1. A properly designed and supervised resistance training program is safe for children
2. A properly designed and supervised resistance training program can increase the strength of children
3. A properly designed and supervised resistance training program can help to enhance the motor fitness skills and sports performance of children.
4. A properly designed and supervised resistance training program can help to prevent injuries in youth sports and recreational activities.
5. A properly designed and supervised resistance training program can help to improve the psychosocial well-being of children.
6. A properly designed and supervised resistance training program can enhance the overall health of children."
Key point I believe is the "properly designed and supervised" part. Where early development of good form (technique) can also give them a head start for when they do decide to lift heavy when older, they will not hurt themselves, as the motor pattern for good form is already learnt.
Good points by A.Schwarzenegger re muscle balance, I too cringe...
It is generally also accepted amongst most strength and conditioning coaches that real young kids do not lift heavier than their 10RM.
I have heard people comment on weightlifting for kids being unsafe, common misconception... Weightlifting for ANYONE can be unsafe if done incorrectly. This brings us back to the "properly designed and supervised" comment again... and if done correctly there are large benefits with no more risk than adults in my professional view.
A point that I have heard, and now often use as an example is that when you jump you land with (about) 12 body weights, I am yet to meet a kid (or anyone) who can squat 12 times their body weight, and this is a controlled movement. So do you stop your kid jumping?
Now I am not saying you should go buy your kids wieghts and say "go your hardest son" as yes this would be dangerous. Get someone who knows what they are doing (ie a certified professional of some description) to teach your kids correct form. Spend a year with them at least, so you know your child is safe. The cost of this would be outweighted by trying to correct injuries caused by poor form later on, and then having to re-learn good form later. Again I know I am repeating myself, but want to emphasise the "properly designed and supervised" part.
Good luck.