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 can not 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. There is a 100 XP reward for doing this, as well as the couple's thanks and Dogmeat as a companion, complete with a variety of bite attacks in combat.