That essay has it partly right, but we have to look at the way the Romans were named to understand how female names worked. On how Roman names worked in general, there's a nice answer here by /u/Celebreth.

The answer to this question depends a lot on what we mean by 'early' Roman times. Italy was a bit peculiar regarding naming conventions, since Italians, including Latins, Romans and Etruscans had a naming system that consisted of two parts, a hereditary name and an individual differentiating name. This was in contrast to most other indo-germanic languages at the time, which didn't have such a concept of a strictly hereditary name, Greece is a good example. The two components of the Roman system were the nomen (or nomen gentile), literally just 'the name', which was hereditary, and a praenomen, the 'fore'name, which is functionally close to a first name. We don't know exactly how this system evolved, and neither did the Romans. Priscian wrote that it came from Romans and Sabines combining their names after integrating them into the polity, but that's just a nice story.

What we do know is that both men and women used that naming system, and most women would have had a praenomen, which were mostly just distinguished by having a feminine form. So for Caius, there was Caia, for Decimus Decima, for Sextus, Sexta and so on. Due to the nature of our early evidence, it is hard to get many examples of this, but Mika Kajava has done a great job of collecting all the evidence and his study is worth a read if you're interested in that topic^1. Because the evidence for Roman female praenomina is quite scarce and all our historical sources on early and Republican Rome come from a time where women did only have one name, this has led to a misrepresentation of the situation which you still can find even in historical studies (or essays).

That being said, I can give you some examples here. This inscription from Praeneste/Palestrina records the name of a Caia Caponia, dated to the 3rd century BC. Caia can only be a praenomen in this context. This one, from 3rd century BC Rome, a votive inscription to Magna Mater, was erected by a Tita Varia, daugther of Titus (Titi filia).

The female praenomen never fully fell out of use until the end of the empire, and survived f.e. in the legal fiction of giving the freedman of a woman the filiation Caia libertus, 'freedman of Caia [i.e. a woman]'. But by the 3rd century, these are only isolated examples, and it seems that the female praenomen practically disappeared from the mid of the 4th century BC. There are some reasons for this, one of the important ones is that the praenomen wasn't that good of a personal, individual identifier. There was only a limited pool of praenomina, and the Romans seem to have been very adverse to adding new coinages here. Also, there was a big tendency to give your children the same praenomen as their father, so the first son of Titus Varius would be Titus Varius Titi filius, his daughter Tita Varia Titi filia. The most important name for a Roman, in more than one sense, was the family name, the nomen gentile, and its patrilinear perpetuation. Survival of the personal name was more important than survival of the bloodline, which is why adoption was so common and usually uncontroversial.

Daughters kept their fathers name even at marriage, but they would then usually identify themselves further by the name of their husband, rather than the name of their father. Thus, when Saufia from Praeneste/Palestrina married Marcus, she was called Saufia Marci uxor, Saufia, wife of Marcus, as she does on this inscription on the base of a column. This served well as an identifier to uniquely identify a woman, and played a part in the disappearance of the female praenomen, while the male persisted in use.

So to sum up the first part, during the Republic, woman usually had only one name, their family name in the female form, inherited from their fathers. However, there are variations on this theme. First, some women might be called by nicknames or affectionate names, as when M. Tullius Cicero, who was quite fond of her, called his daughter Tulliola, instead of Tullia. Some families who already used the cognomen, the third part of the canonical Roman male name, such as the Caecilii Metelli, had female members called Caecila Metella as well, and fathers who had more than one daughter would give them cognomina such as Major and Minor, or Tertia, Quarta and Quinta, to differentiate them by order of birth or age.

In any case, during the end of the Republic and the beginning of the Empire, the use of cognomina rapidly rose, first among the upper strata to then be emulated by the lower classes, and by the first century, a cognomen as a further individual and more flexible identifier was widespread and common for men and women alike. This finally also led to the decline of the male praenomen, which fell into disuse.

As for Christianity, I have written about the impact of it on Roman naming conventions here, so I'll just say that it did change some things, mainly the names that were chosen - names with biblical connotations, or names directly referring to christian religious concepts became popular and changed the names women were commonly given. By that time, the Roman naming system had pretty much reverted to a single name system, were people usually would have only one name. The flexible cognomen had proven to be a sufficient method to uniquely identify an individual, and the gentilicium, the family name, gradually disappeared for a few centuries.

1 - Mika Kajava, Roman Female Praenomina: Studies in the Nomenclature of Roman Women (Helsinki 1994).