This is an interesting subject, and one I will watch. I have certified one 10 year old, and when I did, the stipulation I put out for his Dad was that Dad attended class with him. Partly for me to feel comfortable that Dad knew what he was doing, and partly for Dad to have a knowlege of what the boy had been taught. His dad readily agreed, and attended every classroom, pool, and open water session.
Truth be told, the boy picked up most of the skills just as quickly, and in some cases more quickly than anyone else in the class, and when I certified him, I was comfortable that he was and would be a good diver.
With that said, I am not a doctor, and am not qualified to make determinations on what potential physiological effects there could be, though from what I have read, I can't find where any studies etc. have been done, so we should consider the information as speculative. We do know that some of the concerns cited are fact, such as the "hole in the heart" (i.e. connected ventricles that heal slower in some children), and the differences in eustacian tubes. For these reasons, IMHO a child should have a thorough physical prior to diving, with emphasis in areas like these. One point to ponder: I havent been able to find any reports of diving accidents with kids, so that suggests to me that people are tending to use the caution that is needed when kids are involved.
The next thing that I would consider is that Children ARE Children, and due caution should be exercised. You wouldn't let your child ride their bike in traffic, and you would be very cautious and provide vigilant supervision with them riding NEXT to traffic, so diving should inspire the same vigilance. Keep to the depth limits. There are plenty of good dives in shallow water, so find them when you dive with kids, and don't just dive with them, SUPERVISE them. Be honest with yourself when it comes to your skills. If you aren't 110% confident that you can handle any situation that might arise, get yourself that way. Take a rescue class, whatever it takes.
First and foremost, you have to be a responsible parent, and decide whether they are mature enough to handle what they are doing. Diving isn't for every adult either, but adults can decide that for themselves. As parents, sometimes we have to say no, even when we don't want to, especially where a child's safety is involved.
With all of that said, I have an 8 year old daughter that wants to become certified. When she turns 10, I will consider it, and if I feel that she is mature enough, I will grant her request and I will most definitely "practice what I preach"