Rome lasted a long, long, long time. Even within countries as young as the US names go in and out of fashion all the time. Names like Aggie or Gertrude that were relatively common just a century ago are kinda weird sounding today. No doubt Rome, in its long history, has had similar shifts in the popularity of names. What were some of those shifts? And, similarly, are there names that were just classics, which remained common throughout all the eras of Rome?
So any answer to this question is going to have a lot of caveats attached, the primary one being that we really don't have a very good snapshot of what most people's names looked like in Roman culture. We have texts and inscriptions that mention the full names of several thousand distinct people (estimates vary, but my recollection is that the RE puts the number at around ten thousand), the vast majority of whom came from the upper echelons of society. In some periods the names are almost exclusively patrician names, and we have more total names in the late republic and empire than in the early republic. So our view of the commonality or lack of commonality of names is going to be skewed by the fact that we're looking at a small slice of people with a specific class composition, and we don't have similar quantities in all periods.
That said, a male Latin name typically had three parts (but sometimes had more), two of which were usually hereditary, and only of one of which was actually chosen by the parents for an individual child. Since you're asking about modern first names, the closest equivalent for Romans was the chosen name or Praenomen (forename). Compared to modern first names, however, there was very little variation in praenomina
There are between 50 and 60 attested praenomina (depending on how certain variants are counted), but many were very uncommon. In the earlier part of the republic, there were about three dozen praenomina that saw regular use. Several of those were almost exclusively attested in single family lines (for example, Appius, which, to my knowledge appeared exclusively in the gens Claudia, but appeared frequently there for hundreds of years).
Certain praenomina became less common over time. For example Hostus was out of use by the 1st Century BCE, and the fact that it had fallen out of use was noted in contemporary sources (i.e. Varro). Similarly Caeso (or Kaeso) was never very common outside of certain families, but fell out of use entirely in the 1st century CE.
However, upper class Romans preferred to reuse family names: typically the first son was given the same name as his father, and succeeding sons were usually given close family names or numerical names, like Quintus (fifth). As a result, new praenomina did not arise to replace the ones that fell out of use, such that the pool of male praenomina in use dwindled over time. A dozen or so remained popular through the end of the western empire, and a few of those actually survived the fall of the western empire and are still in common use: Marcus and Lucius and their direct adaptations (Mark, Luke, Lucien, etc.) are collectively still fairly common names. One of the most consistently popular Roman names, Gaius, survived in derivatives in Romance languages, but is not common in English-speaking countries.
I'm separated from most of my reference library, so I'm working largely from memory, primary sources, and a copy of Munzer's Roman Aristocratic Parties and Families, which briefly addresses nomenclatural problems. The RE (the Realencyclopädie der classischen Altertumswissenschaft), which I do not have on hand, has an excellent article on the subject if you want to learn more. When I'm near my library again I'll try to fill in more sources here.