I would second most of the above. We have always had a house rabbit or two ... my daughter breeds show rabbits, and there always seems to be a kit who moves into the house for some reason and then never leaves.
We currently have two in the house.
We have 3 dogs. Two of them are fine with the rabbits. The third one we have to watch ... not because she is aggressive, but because she tries to play with the rabbits, and there is no way to explain to her that a 45 pound dog can do some damage. You do have to watch your electrical cords, as well as steps or anything the rabbit can fall off of. Our rabbits stay in their cages unless someone has the time and inclination to supervise them. They are out every day, but they do not have the run of the house 24/7. Rabbits are a little more fragile than cats or dogs, although I wouldn't agree with the idea that a pet rabbit is going to drop dead just by being startled. You will not have to worry about the rabbit every time the dog barks or a door gets slammed. They get used to the activity in your house. Injury is really the thing you have to worry about.
With a 6 year old, it has to be understood that any pet is ultimately the responsibility of the parent. A 6 year old is not developmentally mature
enough to have total responsibility for anything.
Having said all of that, a rabbit might actually be a good choice for a child who is normally gentle with animals. I would suggest looking for a Netherland Dwarf ... this is what my daughter breeds. They weigh about 2 lbs. at maturity, which is a good size for a child. Most pet rabbits are injured because the rabbit is too big for the child to pick up, and the child ends up dropping it. ND's are generally calm and sweet tempered and make excellent pets. My daughter has sold a lot of pet quality rabbits to families with children. Our rabbits have never seen a vet ... my daughter is knowledgeable enough to take care of most problems on her own, and to be honest, by the time you know a rabbit is sick there is often little you can do for it. Rabbits, like most prey animals, are experts at being absolutely fine until suddenly, they're not. We have never lost a rabbit to injury ... when my kids were smaller, they were only permitted to hold a rabbit if they were sitting on the floor. That way there was nowhere for the rabbit to fall if the kid lost his grip.
So I've said all that to say that like any pet, you need to understand the pros and cons and then make the right choice for your family. Personally, I think small rabbits are good first pets for most kids.