Ask yourself, what pet do you want? What amount of work are you willing to undertake? Because in reality, while your child may help, you are going to be doing the work (if the pet has an accident, or needs it's cage cleaned, you or your DH will be doing that work; if the animal needs to go to the vet for whatever reason, it's your money & time, not the child's).
That being said, we had two large (65# & 75#) dogs, ages 7 and 8 when we had our son. They have been great with our son. But a lot of that is a) their personalities [aka, forgiving & laid back], b) they are well trained [sit, down, stay, bed], and c) we do not allow the dogs to be alone with our son. They dogs, and it's not fair to them or the child to leave them unsupervised with a child, even a well behaved child. When we got them, we put them both in obedience class, at a cost of $75 per dog. We've had routine vet care for them except for 2 minor issues years ago, at a cost of about $400 (includes vaccinations, flea stuff to keep fleas away & heart worm prevention, total for both dogs). However, our older dog just had to have a cancerous tumor removed, at a cost of $600. We also worked with them a lot, when we first got them, to train them (a lot was 10-15 min per day, and then repetition of the commands throughout the day.) Other equipment (crate for when we're away or at night, $100 + $100 for miscellaneous other stuff like pet beds, leashes, collars, ID tags). Also, we have a fenced yard, which has been a total lifesaver (not as healthy for me, because I don't take them for walks much any more, but so easy in terms of taking care of them). We did hire a professional pooper-scooper service while I was pregnant & the baby was an infant, and that cost $500/year. But I do that work now (again); it would be my least favorite aspect of pet ownership.
There is no way I'd get a puppy at this stage--it would be too much work and stress on me (I'm the "primary caregiver" for the pets, lol). If I were in your shoes and I wanted to get a dog (I personally think dogs are great, better than cats, for little kids), I would suggest going to a humane society or other rescue and look into adopting an adult dog (age 3 or older). Ask them if their dogs are temperament tested (so that they can tell you if the dog is skittish or not, rambunctious or not, food motivated, etc.). Also get a book from the library on dog breeds, so you can see the various attributes (for example, Golden Retrievers are wonderful family dogs, but have long hair, which can be a pain).
Or, consider adopting an adult cat (rather than a kitten), one that has been temperament tested. Cats are much less forgiving of human's mistakes, imo (I love cats, mind you and had one till he passed at age 19). But some cats are great family pets.
For me, personally, if we're going to have a pet, I want a pet that I can interact & bond with, if I'm going to be doing the work of caregiving for it. Which is why we don't have fish, turtles, etc. I'm not saying that I couldn't bond with them, but it's easier for me to bond w/ dogs or cats then fish or reptiles. Down the road, when my son is older (maybe 9 or 10), we'll consider another pet of his choice, maybe, depending on how responsible he's proven to be with his stuff and with the family pets.