The search for his father represents Shmuel's heroic quest in The Boy in the Striped Pajamas.
All heroic quests begin with an important mission. Shmuel divulges this when he tells Bruno that he must find his father. Shmuel tells Bruno that his father "went on work duty with some other men and none of them have come back." Shmuel further informs Bruno that while he did "some exploration" to try to find him, he was not successful.
Like many heroic quests, Shmuel enlists help on his mission. When Bruno proposes to cross the fence into "Out-With," Shmuel recognizes that he could have assistance. Shmuel recognizes the value of having "someone to help him in the search for his papa." When Bruno crosses into "Out-With," he does not like what he sees and wants to return home. As with many archetypal heroic voyages, the hero's focus does not deviate despite calls of protest or resistance. When Bruno says he wants to stop looking, Shmuel reminds Bruno of the mission at hand: "Shmuel stopped walking and starred at him. 'But Papa,' he said, 'You said you'd help me find him." Upon being reminded of his promise and the importance of the quest, Bruno agrees and both boys continue to search until they are no longer able to do so.
While it might not have been intentional, Shmuel sacrifices his life for the quest. This shows a hero who gives everything they have to fulfill their quest.