Benjamin (בִּנְיָמִין) is a Hebrew name that means "son of the right [hand]." The shortened form Ben is the Hebrew word for son (בֵּן). While the character in National Treasure is named after Benjamin Franklin, the etymological meaning of his name coincidentally illustrates the filial obligation he feels, not only to his ancestors but also to the Founding Fathers. This is evident in the pledge that the grandfather devises for Ben early in the movie: "Benjamin Franklin Gates, you take upon yourself the duty of the Templars, the Freemasons, and the family Gates." In light of his name, it's also appropriate that the search for the treasure culminates in him, the son of his forefathers. As Patrick says after they've found the treasure: "We're in the company of some of the most brilliant minds in history because you found what they left behind for us to find and understood the meaning of it. You did it, Ben. For all of us. Your grandfather and all of us."
Although it doesn't fit as well as Ben's name, Patrick's name also provides a hint of the father-son relationships in the movie. The name itself means "noble," but it resembles patris, the genitive form of pater, the Latin word for father.