There are several Byzantine threads already present on this sub, but none of them (that I read at least) could specifically answer this question. So, just what I asked in the title! In my own research, I have overwhelmingly only been able to find the names of imperial and noble families. Was the Byzantine Empire in that time period of civilization where people of lower station still didn't really have surnames, or used patronymics?
Just to be clear, there was a roughly 200 year period between the Arab Conquest and the Macedonian Dynasty where even the noble families did not have surnames (or rather, there were none that were trackable across the divide). In that time period, aristocratic identity ceased to mean much in the face of existential military threat, which demanded a more meritocratic system based around the state and the military itself.
I talk about it a lot in this link.
The comment from the JSTOR article at the end would imply that once surnames began to come into fashion by around the 9th and 10th century, any peasant adoption would follow the same form, being descriptives of location, occupation, or direct ancestry (rather than ancient ancestry).