I only skimmed over this long thread, but I've done a research paper on the byzantine empire, and think I can give some insight on the byzantine empires strengths and weaknesses.
First about the recent posts about the Cataphract. It should replace the knight without a doubt. It was not considered heavily armored compared to light horsemen, oh no no no. It was considered the most heavily armored horseman in history. The downfall of the cataphract to the knight had more to do with the empires decline rather then the superiority of the knight. Compared to knights the cataphractoi would be slower, but rather then giving them a bonus to attack strength It might be interesting to portray their superior armor with cover promotion, and portray their armor-piercing maces with a bonus against mounted units. Cataphractoi and knights may have been similiar, but they were used completely differently. Compared to knights cataphractoi were more like tank-like and slow. Probably too slow to bother with flanking, but armored enough not to care.
Instead of the cataphractoi they could also put in the varangian Guard. Fearsome viking mercenaries who had full access to the royal armory. Imagine the fearsome viking soldier with his two-handed axe, and fighting skills. Now imagine that viking warrior with the most advanced armor, and weapons available at that time period. It must have been a sight to see.
The byzantine Empire's biggest strengths came from its professional army, its extensive use of mercenaries, and Constantinople.
professional army
While the other nations relied on feudal armies consisting of poorly armed militias combined with a few elite noblemen, the byzantines had a professional army. he decline of the byzantine empire started when they got rid of this professional army. Maybe in the game this could be portrayed as a reduced maintenance cost for military units, or maybe an increased supply.
Mercenaries
Any gaps in the byzantine army was filled in with mercenaries. In battles its safe to say the majority of their troops were likely mercenaries, and dang good ones at that. Turkish bows, hungarian horsemen, viking axes. Anyone and everyone. In the game I have no idea how you would portray this advantage.
Constantinople
There is so much that could be said about Constantinople. No city could match its defenses, nor its wealth. It had great aqueducts to support its large population. Everyone who saw this city would have to wonder if god himself built it. If you want the byzantine empire to play like the byzantine empire we all know and love their UA should probably focus on capital city advantages. Growth bonuses, trade bonuses, defense bonuses. In the game the byzantine empire should always have a powerful capital city.
Right. I hope this gives you all some more ideas, even though they probably already came up with their decision, and if its anything like giving America minute men, despite the fact that America's standing army lost the majority of their battles, I'm going to spit.