This website has some really good information on Roman naming systems. Basically, Romans had a praenomen, a nomen and a cognomen - given, clan and family names, the latter of which usually originated as a nickname based on the personality or appearance of the founding member of that family within the clan. Even a Roman citizen might not necessarily have a family name, or they might have several names. The linked website also suggests some suitable names for each category. It appears to be targetted at users of the website who are, it seems, required to pick a Roman user name, but it's useful for research too.
Romans could also EARN additional names depending on their abilities. The dictator Sulla, for example, was called Lucius Cornelius Sulla Felix. He earned 'Felix', which means 'fortunate' or 'lucky', in his later military and political career.
If you're 'noble', then you're more likely to have multiple names. The first name is your birth name, the second your family name and the third your branch of the family. Any additional names would be personal 'nicknames'.
If you're not noble, you might have only one or two names.
The main character of my NiP is a former centurion by the name of Secundus Duro.