Personally, this might not be popular with a lot of people, but I recommend age 5 at the minimum. Maybe even older depending on the maturity level of your child. Puppies need a lot more work than older dogs, since they need to be housetrained and supervised constantly. They often do a lot of jumping up and play-biting/mouthing, and that takes additional training to correct. Usually by age 5, most kids don't need constant supervision and attention so it makes taking care of the puppy easier. Also, at that age kids usually can be expected to have a better understanding of how to behave appropriately around a dog and the rules that need to be followed - no pulling ears, poking with sticks, taking toys away, getting in the dog's face while it's eating, running, screaming and chasing, etc. There are always exceptions - maybe a 3 year old would do okay with a calm, well-behaved, child-friendly adult dog - but in general I would wait until at least age 5.
In my experience people seem to think they need to get a puppy when the kid is 1 or 2 so they can "grow up together" but waiting until your child is 5 or 6, and getting a dog that might live as long as 15 years, will still allow that to happen.
Just for the record, I had my 2 dogs before I had my daughter. The dogs are now 8 and 13 (both pugs, a very family-friendly child-tolerant breed) and my DD is 3. She's always been good with them and seemed to understand from day 1 how to be gentle around them.