To start this quest, the Vault Dweller needs to go to the Casino area of Junktown. To the right of the entrance area, there is a pair of people trying, unsuccessfully, to enter their house. The reason they cannot enter is that Dogmeat is blocking the door.
Simply wearing a leather jacket (which will remind the dog of its former master) or giving the pooch some iguana-on-a-stick will solve the quest and let the couple enter their house.
In the Fallout Demo, Dogmeat didn't play a role in this quest. A unique dog, simply named "Dog," was found here instead, and the only option to get it as a companion was to feed it iguana bits (which appear as "on-a-stick").