Child has swollen gum in roof of mouth after tooth extraction. What is the best remedy?

Call the dentist. To find out if this was expected and how best to handle it, or if it requires that you child be seen again.
Salt Water Rinse. If the swelling is right in the center of the spot where he tooth came out then it is most likely a blob of gum that had grown up inside the baby tooth. This should resolve on its own with in a few days. Warm salt water rinses could help by keeping it clean.