The Byzantines have had remarkable stability which is a strength for the civilization and remains somewhat viable due to an excellent flow and synergy of bonuses. While it may be a bit soon, perhaps we should look at tuning them to face what are becoming potent competitors.
The Byzantines largely rely on excellent counter units to work their way up to a cheap and powerful Imperial Age. Historically they lack Blast Furnace, and Bloodlines, but in turn have fairly versatile unit spread due to their Camels trading cost effectively, and their fully upgraded Arbalesters. These may not be enough with some of the upgrades civilizations are getting.
My proposition is:
Counter unit upgrades cost 40% less and research 50% faster (Including Cataphracts). They are trained 5% faster.
Cataphracts are now a Counter unit for the discount, based on the original Cataphract price.
The accuracy of Fireships is adjusted to the increased range when Greek Fire is researched. Ships, buildings and siege units hit by them cannot be repaired for 5 real time seconds. The ship will also move 5% faster.
Greek Fire adds small area of effect damage to Archer lines (Foot and CA). Ships, buildings and siege units hit by them cannot be repaired for 2 real time seconds.
I avoided giving the Byzantine Romans Blast Furnace, which would have strengthened their Halbs and Camels to maintain the original feel. Giving them Bloodlines really is not anathema and enables them to fight off the more potent cavalry that is running about at this time. Some civilizations will find this difficult to compete with.
Lowering the cost of Hand Cannoneers is another choice in place of the Cataphracts but I think it might be well too much. Both are supposed to be difficult transitions. (Cataphracts may have to have their base HP lowered by 10 to make this fair)
The choice to have it affect Archers was a reference to handheld Greek Fire. It can simply be countered through formation play. The area of effect is not large, but it could be useful. Area of effect damage will be low.
I really do not care if they get the Steppe Lancer.
Allowing Greek Fire to affect the ability to repair will transform the technology significantly. I do not foresee a need to change the price.