Corroborating Anestis Samourkasidis’ answer: Vote #1 Anestis Samourkasidis’ answer to When did Greeks as a people adopt surnames?.
If you peruse Prosopography of the Byzantine Empire part I, you will see sporadic surnames in the 7th and 8th century; e.g. I PBE: Ioannes 9 : Ἰωάννῃ σπαθαρίῳ, τὸ ἐπίκλην Στρούθῳ “John the spatharios [military office], nickname/surname Strouthos”. Surnames were still rare, and you’ll notice that PBE I doesn’t even use surnames in its index (hence 607 Johns in its database). Certainly by the 12th century, surnames were universal.