Now, this isn't exactly what you were going for, but I can give you a piece of the Americas. It's quite achievable in any Byzantine wank, even those built to be quite plausible.
Now, I doubt they'd be the ones discovering the Americas, especially looking for China. For a resurgent Rhomaion, there's no reason to circumvent Africa or vastly underestimate distances and head west looking to avoid the middle man in trade. That's mainly because they are in the middle man for Chinese and other eastern goods to Europe.
They're also much less likely to go on a colonizing spree in the New World upon its discovery by a western maritime nation, as if they've gone to the task of retaking their previous land (i.e.: Egypt), the East Indies is a much safer objective for Roman colonies with a known profit to be had. Not only will the early years of Colonization be without secure returns, but the Romans could be quite swiftly cut off from their own American colonies by its competitors. They might be kings of the Mediterranean, but a large Atlantic fleet would strain supply lines and likely be easy pickings for the likes of Spain, or England, etc...
It'd cost a lot of manpower to maintain, where the Romans just can't compete with the Western European powers also in the Americas while split between the East and West Indies. Which is exactly why they wouldn't focus on the New World at the end of the day, despite the possibility of all those other negative factors: It takes away part of their ability to get a larger share in the East Indies, their irrestible desire. But still, I did promise a bit of America.
Your answer is... *Drum roll*... some islands in the Lesser Antilles! Not great, I know, but it wouldn't be incredibly difficult to get ahold of several. Quite a few powers had their fingers in those islands, even those who were not particularly large on colonial efforts. It'd mostly be a prestige acquisition for the Basileus, or Senate, the leaders in whatever kind of Byzantine wank we're in, but it's a neat little bit I've heard proposed. Basileus444 (whom I won't @ without permission), author of Age of Miracles, floated the vague idea a few times when readers would ask the same question as you.
I personally find it an acceptable prize for the Byzantines. The only way I could see full out colonization à la New England happening is in a Byzantine wank pumped so full of steroids that even the empire itself just wants to die. Still, that's just my opinion, and what's my opinion!?
Dumb, that's what it is....
Curious to see what others are thinking though.