Greek Naming traditions, 1st C BC -- 1st C AD

[M. Demetrius]

Had the Greeks adopted Roman naming traditions by this time, or was there some other system for naming sons and daughters in a rural Greek family?

[Epictetus]

No, they didn't whole-heartedly adopt Roman names, or the structure of naming.

It was usual for the Romans, in the person of a conquering general, later the emperor or a governor of a province, to make individual grants of Roman citizenship, which involved the recipient in adopting the Roman style of nomenclature. The Roman naming system was fundamentally different from the Greek, involving three names, praenomen, nomen and cognomen, with the patronymic expressed by a formula which came after the nomen. Thus: G(aius) Julius G(aii) f(ilius) Caesar.

Greeks receiving Roman citizenship would adopt the praenomen and nomen of the donor; in this way, names like Julius and Flavius first made their way into Greek nomenclature, in a variety of spellings (Φλαουιος, Φλαβιος ). For their cognomen, though Greeks did sometimes adopt Roman names, they more usually retained their Greek name. Thus, Gaius Julius Alexander, Titus Flavius Alcibiades.

The traditional Greek patronymic system did not fit easily into the Roman pattern, and Greeks adopted a number of strategies to cope with the problem. Sometimes they imitated the Roman system completely, but sometimes they simply appended the name of the father after the cognomen, either simply in the genitive, or with an ‘explanatory’ formula. For example, Τιβ. Κλαυδιος Αλεξανδρος πριν Φιλιππου ‘Tib. Klaudios Alexander, formerly son of Filippos’, meaning not that the person had been adopted, but that previously he would have expressed his patronymic this way.

Lexicon of Greek Personal Names

This is a really good site, by the way. You can even search by area to see what the most popular names were.

Edit: Here is a page on naming practices.