Running Shoes Are Kind of Like Children...

For starters, running shoes DO make a difference. Years ago I made the same mistake that you may have made in thinking that any running shoe from a big company will be fine. You end up shopping in the clearance section and at big sporting goods store while trusting a shoe salesman that doesn't even run or perhaps runs 5 or 6 miles a week. I'll save you ten + years of trial and error by sharing the following.

• I say this jokingly but it helps illustrate a point. Running shoes are like children, they all have different strengths and qualities. Knowing how to use those qualities allow you to enjoy the shoe much more. 
• Shop for running shoes at a place where avid runners are the sales people. The shoes might be a little more expensive up front but because you are more likely to get into the right shoe it is cheaper in the long run. It's cheaper because you avoid injury and also a trip to the store to buy a better pair.
• Start your child off with shoes that promote a proper stride.  Saucony Kinvara has youth and adult sizes. Altra running shoes will soon have youth sizes and adult shoes cut specifically for men and women feet. Other quality brands with minimalist shoes are New Balance and Brooks.
• A running shoe lasts for years, but it should only be ran on for the first 300-600 miles (your weight and running form determine where you fall on the spectrum). If your child is wearing their shoes to school and playing around in them after school you need to factor in those "miles" as well. If you're not into factoring things out you can go by a general rule of thumb that children under 12 should replace their running shoes each year, teens in Jr. High and High School should be getting new running shoes every six months of running.
• If you have a child in high-school it is wise to have two pairs of running shoes (different brands) they rotate every other run. This provides time for each shoe to decompress and also provide the foot with a little different feel each run so repetitive injuries are avoided.

Although running shoes are like children they don't get better with age. If you want your child to be serious with running you'll need to get serious about shoes. Shoes will be your most expensive item in this sport, but if you take the time to find the right shoe they will keep your child running rather than sidelined with injury.