Why do peasant characters in 19th century Russian literature so often have Greek names? (e.g. "Agafon" and "Platon" in Anna Karenina).
Firstly, why there are Greek names in Russia. Russia, being a Christian Orthodox country, had strong historical and cultural connections with Greece. So, many Russian names are of Greek origin. Most of them are archaic nowadays, but some are very common.
Secondly, in the XIX century, there was a somewhat strong distinction between "noble" names and "plebs'" names. For example, when introducing Tatiana in "Eugene Onegin" Pushkin adds a note:
The sweetest-sounding Greek names, such as Agathon, Philat, Theodora, Thekla, are used among us only by the common people
Yuri Lotman (referencing Nikonov, M. 1974) states that that time the name Tatiana was given to 18-30 peasants girls, and only to 10 nobles out of 1000.
Thirdly, why were common people more likely to have Greek names. For me, the most probable explanation is that than peasants were very religious and usually named their children according to the Menologium, after Orthodox saints. And saints usually had Greek names. On the other hand, the aristocracy often had different ways to choose a name, fashion trends, for example.
Fourthly, Tolstoy references Platon (the philosopher) in Anna Karenina, so he could choose that name for a character for some symbolic or creative reasons.