Why would a puppies belly button swell?

Might be a hernia. take it to a veterinarian to be checked out.
That sounds like a hernia to me. You need to take your dog in to be checked out by your Veterinarian. It can either be repaired surgically or possibly reduced back in and taped down but if it is an older dog and not a puppy, surgery would be the only solution but first you need to see a Vet to make sure what it is.