I've always wondered about this (particularly with ancient Roman/Latin names, as I'm into that stuff).
When a name means something like "green hill" is that exactly what it sounds like to a speaker of that language?
Like this is my daughter, Green Hill Jackson - sounds weird to me.
I know back in the Roman times, it's like everyone and their brother was Gaius (at least in the earlier years), whereas there could be quite a few family names. Did those come about in the same way? Do they have that literal sound to them, or do they just seem like identifiers the way we do now? (like how when someone hears my name "Christopher" they don't think of "Christ-bearer"?)
Toponyms stack up over time, particularly when a group which speaks a different language takes over a bit of land. Thus, the place that used to be called Tor (hill) by the locals becomes Tor Pen (Tor Hill), then Torpen How (Torpen Hill), and finally it's modern name Torpenhow Hill, or literally HillHillHill Hill. This is actually a great way for historians to see who ruled where, and who conquered whom.
Personal names are a bit more tricky. Yes, many Christian names are compounds which have lost their original meaning. Peter famously means rock (lat: petra), for example. There is also a marked tendency in early Christianity to take appropriate baptismal names. The Gospel of Luke and the Acts of the Apostles are both addressed to "Theophilus", which literally means "God-Lover" in Greek. Whether this is actually an individual or not is the subject of debate, but the name is attested in Pompeii graffiti. The names have obviously persisted while their original meaning was lost as Latin became Romance, and Greek was lost in the West.
I say tricky because you mentioned Rome. While there were a very small number of first names, people tended to pick up cognomen or nicknames. Cicero, for example, the cognomen of Marcus of the Tullies (Marcus Tullius), means 'chickpea'. So, just because the name comes first doesn't mean you can assume it was used in the same way.