There's a term for them, actually, they're called patronyms. I myself have one that many people don't realize is a patronym; Mackenzie. Mac = son of. You see them in a lot of different cultures, particularly Celtic and Scandinavian.

Edit: realized I didn't quite answer you. Having a surname isn't something that is universal to every culture. It's grown in the modern world but many people took names based on their profession, (Smith, Tailor, etc), a patronym or clan name, their hometown, or just aspirational (King, Duke, etc).

I heard a story once, not sure if it was true, that in Mongolia, an attempt to modernize some government system or other prompted them to ask the population to just choose surnames for themselves, and it backfired because a huge percentage of people picked Khan.

Essentially, patronyms emerged out of basic heredity, like father like son. I'm the son of Coinneach, call me MacCoinneach. I'm the son of Vlad Dracul, call me Dracula. Romans had cognomens, like Caesar, which were sort of hereditary nicknames.