That depends on the time period and location. Up until around the sixth century, you can see a lot of emperors having Latinate names, like Flavius Sabbatius Iustinianus (not that I'm biased!), Constantinus, or Iulianus. Later, more Hellenic names appear, like Basileios the Bulgar-Slayer or Alexios Komnenos. It's worth noting that those two are usually anglicized and latinized, respectively, as Basil and Alexius Comnenus. Now, since the Byzantine heartland was around the southern Balkans and Anatolia, Greek names would dominate the lower classes in much of the Empire. At its further reaches, one could also find Slavs, Armenians, Arabs, and others. Furthermore, in Constantinople there existed a large Italian presence thanks to the merchant republics.

Again, though, you're asking about a period of roughly a thousand years and an area covering much of the eastern Mediterranean. It's hard to generalize.

Follow up, when did the Byzantines stop using Latin and changed to more Greek names?

Background (to build upon /u/EMPEROR_JUSTINIAN_I) Historically Hellenes practiced mononomia (meaning they only had one given name, i.e. Alexander or Pericles). Typically those names would have two components just like Germanic names. In addition with the long recorded history of Greeks and Hellenes there was a wide record of possible names to give to your child. After the Roman conquest, Greek Roman citizens use a mix of the Roman trianomia (usually the praenomen and nomen of their first sponsor) using their Greek name as cognomen. With the declining use of the trianomina among Roman citizen starting the 4th century BC, Greeks simply reverted back to mononomia in their own language. By the 6th century, even in Latin record of the ERE most individual are only identified by their given name, only old senatorial families keep alive the practice of multinomina.

One of the most important factor in the change of naming conventions among Hellenes is obviously Christianity bringing both names from the Bible, and names related to Christian concept or affiliation (i.e. Theophilos or Christophoros). Side note: some major Latin names made their way into Greek (most notably Constantine), while Greek gave many Christian name to Latin speakers.

So what about the 13th century: If you look at the Palaiologos line, you see names heavily influence by Christianity, both names from the Bible and theophoric names. But some classic Greek names remain too (Andronicus in this case). This is a feat that is also present in the west with Germanic names coexisting along side Latin/Christian names in the same family (the Capet dynasty of France mostly used Charles, Louis, Robert for their sons since the 10th century, names which are all Germanic but later they also started to used Philippe (i.e. Philip) a very old Greek name, and later Jean (i.e. John)). The 13th century is also a complicated time for the Empire falling to the Latin in 1204. So it is very likely that at some level Germanic or Latin names penetrated into the Greek culture among the lower classes at least for a time. Another interesting things is the use of several family names in the mobility to demonstrate lineage (especially in relation to the former imperial lines), when the West was only starting to spread the use of family names for the commoners.