When you think of ancient names, do you think of Romans with multiple names like Gaius Julius Caesar, but of Greeks with single names like Plato, Aristotle, or Pericles? There is a good reason for that. It is thought that most Indo-Europeans had single names, with no idea of an inheritable family name. The Romans were exceptional.
If you want to find out more, read about the naming conventions from Athens through the Roman Republic here.