There were, in fact, somewhat around 18 names commonly used in total.
Let me see if I can get a list…
Aulus
Appius
Gaius (Caius)
Gnaeus
Decimus
Kaeso
Lucius
Marcus
Manius
Mamercus
Numerius
Publius
Quintus
Servius
Sextus
Spurius
Titus
Tiberius
Even better, women were usually just known as their family’s cognomen, ie. ‘Julia’.