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").
Behind the scenes
The following is based on unverified behind the scenes information and may be inaccurate.
The quest is a reference to the movie Mad Max 2 (aka The Road Warrior). In the movie, Max wears a black leather jacket and is accompanied by a dog that strongly resembles Dogmeat.
Phil describes Dogmeat's previous owner as "a tall, dark-haired guy with a little graying around the edges, dressed all in black leather, and carrying a shotgun." This description matches all of Max's characteristics in Mad Max 2.
End of information based on unverified behind the scenes information.