With respect to classical Greek names, direct theonymy was exceedingly rare yet definitely attested. This can be easily seen by searching on the Lexicon of Greek Personal Names (LGPN); we can see that, while some gods are not attested in theonymy (Zeus and Hades, for instance), other gods and goddesses are actually quite well-represented, if in very small numbers. (Artemis could be noted as a relatively common name, numbering 63 times in the data base.) It is also worth noting that this practice seems to have started among peoples outside the mainstream Greek poleis, but it did become an established practice, albeit still uncommon, at least by the Roman Empire. (If we are speaking about post-classical times, there are a rather large number of Greeks today who bear direct theonyms of classical pagan gods, like Aphrodite or Athena. I am not familiar with the origin of this practice, but I would ascribe it to modern Greek nationalism if I had to guess.)

Rather more common are names that are derived from a theonym. Greek names were typically compounds of two parts, and it is quite common to see one part of the compound being a theonym: Ἀπολλόδωρος Apollodoros "Apollo-given", for instance, or Διογενής Diogenes "Zeus-sprung" (originally an epic epithet of kings, i.e. "whose rule is upheld by Zeus"; e.g. διογενὴς Ὀδυσεύς "Odysseus the Zeus-ordained"). This sort of theophoric naming seems rather less restricted than direct theonymy; and deities whose names could be used include both major gods, natural powers, personifying deities and heroes (somewhat rare in this last category and really mostly Herakles). It is also worth noting that θεός theos "god" also has its own derived names, some of which still survive today in English like Theodore. (This particular name predates Christianity, although it was certainly considered to refer to the Christian God by later Christians.)

In Rome, the situation was a bit different. Romans famously had the tria nomina system: a three-part name (which was originally a two-part name plus a nickname that some people had; others had even more names given to them due to victories or adoption) in which the given name (praenomen) is drawn from a vanishingly small pool. This is why most classical Romans had as given name one of Marcus, Quintus, Gaius, Publius, Lucius, Gnaeus, Titus, Tiberius, Sextus or Septimus (plus a number of other less-used ones). Most of these names have obscure origin except for the numeric ones (Quintus, Sextus and Septimus), but we believe that Marcus in particular derives from the theonym Mars. The number of nomina was also not great, and most of them don't seem to be obviously theophoric (except, well, Marcius, being related to Marcus, if the latter is theophoric). Direct theonymy is unheard of in these cases, at least in classical times.

As for the cognomen, many Romans of Greek origin chose to use their Greek name in this capacity, so there are obviously theophoric names of Greek origin. Native Latin theophoric names include e.g. Saturninus, adj. of Saturnus (Saturn). Direct theonymy of native Latin names was, again, extremely rare, but by late antiquity we could find examples, mostly Christians such as Saint Mercurius. (Of course, there were also deified mortals whose names automatically become theonyms, such as Caesar, but this probably isn't the kind of name you were thinking about.)

Source: Robert Parker, "Theophoric Names and the History of Greek Religion"