Kids can make great dog trainers. It's just a matter of starting them off on the right foot, using the right methods and always remembering that the experience is supposed to be fun for your child and for your dog. That means that you need to use a training method which rewards the dog for doing the right thing, but which does nothing at all when the dog goofs. The last thing we want to do is to teach our children that it's okay to punish another being for making a mistake.
Clicker Training is the most effective and gentlest training method available. It was developed in the 1960's and 70's by dolphin trainers and is now widely used at zoos and aquariums all over the world to teach marine mammals, fish, and zoo animals. Over the past decade, Clicker Training has gained huge popularity with pet trainers - and not just for dogs. I've used it with cats, mice, hamsters and tropical fish. And I've seen it used to teach a pet rabbit to play basketball and to teach a chicken to run an obstacle course!
There are lots of sources out there to get you started and your child started with clicker training. ClickerTraining.com is run by Karen Pryor, the patron saint of pet clicker trainers and has a wealth of articles to get you started. Karen's book Don't Shoot the Dog is a modern classic among teachers of both humans and animals. The book being shelved in the Psychology section (not the dog training section) of most libraries says a lot. There is also an excellent set of DVDs called Clicker Puppy aimed specifically at teaching children to train young dogs.
Check out any of those sources, then set aside five or ten minutes a few times a week to work with your child and your dog. In days you will see a sense of pride and accomplishment in both your dog and in your child.