I always thought that Roman cognomina (is that even correct?) ending in '-anus' or '-ianus' meant that the man had originally come from one gens, but was adopted by another. The most famous example I know is the Emperor Augustus, born as Gaius Octavius, who was called Gaius Julius Octavianus after adoption by Julius Caesar.
Reading about the Flavian emperors, though, I don't see the same pattern:
First, we have Vespasian, born as Titus Flavius Vespasianus, but not originally from any Vespasius family. 'Vespasianus' seems to come from his mother's name, Vespasia Polla.
Second, we have Titus, so known to distinguish him from his father Vespasian, since they have the same birth name, Titus Flavius Vespasianus. Okay, that makes sense - lots of Romans have the same name as their father.
Third, though, we have Vespasian's second son Domitian, born as Titus Flavius Domitianus. Why Domitianus? It seems to come from his mother Domitilla. But wait, Domitilla was Titus' mother, too. So why is Titus called 'Vespasianus', but Domitian called 'Domitianus'? Why do they have different cognomen at all? Why isn't Domitian instead distinguished with a different praenomen, say hypothetically, Gaius Flavius Vespasianus?
My guesses for why this might be are:
I have the wrong idea about cognomina, and they could have been chosen for a variety of reasons, including simple parental preference.
It was not unusual to take a cognomen based on your mother's name.
Being given the same name as your father was a method of demonstrating that you were his successor. It seems that Titus was definitely Vespasian's "heir," and Domitian was left to the side.
Naming conventions shifted between the 1st Century BC, when Octavian was adopted, and the 1st Century AD, when Vespasian gives his sons two different cognomen.
Many aspects of why individual Romans have certain cognomina are still not well-understood.
Some combination of the above.
Are any of these correct? I appreciate any insight anyone can provide!
EDITED for some grammatical clarity.
This won't be entirely useful, as I'm not a Classicist, but I just read this article on Roman naming conventions and the author actually addressed the Titus/Domitian issue. According to him, because the name 'Titus' (and other praenomina as well) were given to all male members of the family, it became increasingly popular to differentiate children by different cognomina instead, with the mother's name being a popular choice. This process was apparently visible even in Augustus' time, as the infamous Varus' sisters were both commemorated in their sons' names, though in both cases their sons still had different praenomina, suggesting that naming practices changed quite quickly.
Like you, I've only read about the 'standard' Roman naming system, so I'm quite surprised to learn that the system was so flexible!