Type of Dog

I'd say a Beagle! But I'm a bit bias because that's my favorite kind of dog! (I looooove their big, floppy ears!) Are you looking for a puppy or an adult dog? Obviously any puppy will be small but I think an adult beagle could still be good for children. They tend to be small-to-medium in size; small enough for a child to enjoy but big enough not likely to get hurt by a 3 year old.

take care what kind of dog you chose because small children often do things that small breed dogs don't tolerate.Golden Retriever or Labrador are more patient with children but also some small breeds are children friendly.i DON'T suggest you Chihuahua or Pekignese because they can get crazy if the kid grabs them in a bad way and they can be very jealous(trust me i got one :)) my suggestion is Cavalier King Charles Spaniel, Poodle(his hair is also hipoalergenic) or some small terrier(the most energic dog) anyway here is a list of small friendly dog breeds http://dogtime.com/small-dogs-kid-friendly.html

Any type of small dog would be fine. I would actually go for a mixed breed of chiwawa and some other dog, but generally if you don't want a puppy that will grow to be fairly big, sticking with small dogs that are puppies is a good idea.