Why did the use of the family name (Cognomen), as one's surname, diminish and nearly disappear entirely during the 4th - 11th centuries throughout the Eastern Roman Empire?
Brief history of the early existing Roman naming practices of the Principate Period (27 BC - 284 AD)
According to Christopher Andersen (1977), the ancient Greeks, Hebrews, and Romans had surnames. After the fall of the Roman Empire, however, surnames disappeared until the eleventh century.
NOTE: In the text above, Christopher Andersen was strictly referring to the Roman Empire which would have been a predecessor to the Byzantine Empire (Eastern Roman Empire | 330 AD to 1453 AD).
It seems that the diminshing use of the Roman three-name practice (which includes the cognomen as the 3rd name) was primarily due to the influence of early Christian & Greek "naming" traditions.
Personal Names of the Aristocracy in the Roman Empire During the Later Byzantine Era…
Personal names in the Byzantine era of the Roman Empire consisted of a given name followed by one or more surnames. Surnames came in three varieties: inherited family names, patronymics, and by-names.
As Christianity became the dominant (and eventually state) religion, it became popular to use the names of saints instead of the three name practice.
The cognomen (or family name) had begun to disappear as well. With the infusion of Greek culture into the Roman Empire, the use of patronymics ('son of') and by-names (both attributive, such as 'the wise' or 'the short', and descriptive, such as 'of Antioch' or 'the tailor') began to displace inherited surnames. The Greeks did not have as keenly developed a sense of genealogy as did the Romans. The Byzantine era being a blending of the two, the value of hereditary family names declined, and so did their use. Family names are completely missing or extremely rare in documents and seals dated from between the 7th and 10th centuries. Eventually, family names were seen as a quaint custom.
Early Christian and Greek traditions