I also have a child who started diving relatively young, although was a little older at 13 when she first certified. Long story short she did a discover scuba with her best friend while on vacation. Her best friend's mother is also a certified diver. Since the other mother and I were both certified at the time we were allowed to tag along at a distance for their dive. I got a very good sense that my daughter understood the physical risks of diving when as she and her friend where making their way slowly up the anchor line my daughter starts to make these sweeping motions in front of her regulator. She was reminding her friend to "breath" and not to hold her breath as they ascended. Both girls after the dive begged to complete their training while on vacation and left with PADI OW Jr. certification.
Our first dive a few weeks after certification was at Casino Point in Avalon, Catalina Island. I had called the dive shop multiple times to confirm they had gear for a smaller diver and that they had full exposure gear. I was reassured that they had gear and a certified DM. When we arrived they did have a wetsuit small enough for her but had full foot fins and no booties, no hood, and no gloves. Meanwhile the DM was in a dry suit and I had hood, gloves, and booties. Poor girl's lips were almost blue she was so cold after the first dive. An hour later surface interval did not get her sufficiently warm. I could see the apprehension in her eye's about having to get back in the cold water so I thumbed the second dive. To me there was not much to be gained diving was from another dive except reinforce her dislike of cold water diving.
After that dive I told my wife she needed her own gear. She got her own wetsuit, hood, gloves, and booties. Later I found a blow out special online and bought her own BC. The only piece of equipment she doesn't have right now is a regulator.
Since she does not like shore dives I tried taking her on a local dive boat out of Dana Point. Conditions that day weren't great and by the time we got to the dive site we were both queasy. By the time we got in the water I had lost my lunch and she was close. She didn't want to do the second dive so I let her thumb that one. On the drive home after nearby lunch I thought the whole day was a bust. But she surprised me by saying, "thank you Daddy for taking me diving." Wow, the kid actually had a good time and enjoyed her time with Dad. That is special.
These days the requirement for warm water and warm weather diving on the family summer vacation is hers, not mine. She tells my wife when planning summer vacations she is happy to go anywhere warm where she can dive. I of course have to pack all the dive gear to take along with us since after all, someone has to look after her in the water.
So 1+ on DarinJ's points on diving with kids.
1. Each kid is unique. There is no age limit and the parent needs to decide if the kid is old enough and mature enough to dive. Talk to the dive instructor as well and interview as many instructors as it takes to find one with both the patience/temperament to train kids but has experience training kids.
2. Kids need their own gear. Yes, they will grow out of it and you will have to replace it but they need their own gear. That is especially true of smaller kids who may have trouble fitting rental wetsuits and BC's.
3. You have to be patient. The thumb a dive rule in particular needs to be reinforced. Your child has to feel comfortable with you as a buddy to say, "I'm sorry, I don't think diving today is a good idea." It's hard for adults to say that to their friends, so imagine how hard that is for a child to say to a parent. But, if you are diving with kids they have to have that comfort with you.
4. They see the world differently. That is part of the charm. As my later story indicates I thought the day was a bust but she had a memorable day. It doesn't always have to be the greatest dive you ever had to be memorable. Sometimes just spending time with them is enough. I've heard that you show your love by the time you give them. She got a whole morning of Dad without any having to share with her brother or sister. Spend time with your kids, it's how they know you love them.
5. HAVE FUN !!!! That is, after all, why we all dive.