I like the idea of the Canaries islands plan, story flavor could add numerous strategic ways for them to flurish...
Good to know that someone likes it!
-Continue selling ornate relics and other espensive things from Byzantium. Even after moving to Canaria, I'd imagine anything that could be brought would be, and more could later be "liberated"(raided) to exacerbate the trade in luxory goods.
I agree that there would be efforts to try and bring whatever loyalists could, but I'd expect whoever controlled their old homes would be unwilling to allow a mass-depopulation, so whilst small groups of richer families may bring a boat-load, larger groups of them will have difficulty, and poor people will have a harder time still (let alone being able to afford the fare without Byzantine help).
Piracy is a brave option though, because without the right flags, or the protection of Castille and Aragon, any returning pirates will have to sail quite a long way, not that they couldn't, but eventually it'd force the Ottomans to deal with the issue. I'm torn on what their breaking point would be.
-Offering passage to Orthodox Greeks, and eventually all fleeing the Turks, on agreement for an indentured servitude deal
Passage to the Canaries? Free trips are interesting, but again, expensive - so that indentured servitude could be interesting - lots of rent-paying tenants, but I wonder if citizen armies would have to become a thing again, as there are no serfs that can be levied (perhaps that would be one contract of servitude).
-Close proximity to African gold trade
I took a look a this and it seems the Gold trade was primary responded to with salt - which means the Romans may end up being middlemen between Western European salt exporters, and West Africa. With France, Germany, England to name just those accessible without passing the straits of Gibraltar. Could be a powerful boost to the economy.
-Agriculture+ocean wildlife for food, along with native giant lizards which would be an interesting addition, and the island apparently had a high seal population
I think we can see the Imperial Menagerie already

Seals would certainly help make waterproof clothing for the Romans too!
-The islands were however already inhabited, the only estimate I found is at 80000 people, so that may be an issue for the Rhomans. However, with these people also come dogs, pigs, sheep, goats, barley, wheat, and lentils, as well as some degree of infastructure and indigenous plant and geography knowledge.
It depends on the populations that can be extracted from the Morea, and Ottoman Greece - Estimated at roughly 4,500,000 - if even a fraction followed, say 1 in 45? (Admittedly 2% of the population leaving is a lot of people, but it would likely be gradual.) That would be 100,000 people, just from Greece. Anyone else who was particularly disinclined to pay the Jizya, or was convinced by Church leaders, there are a lot of populations to draw from, but it'd take one concerted effort to gather significant numbers. (For perspective England wouldn't exceed this population till the mid-1600s!)
-Would the Byzantines be able to bring along silk production? That could be a great trade boost.
Eh, it had declined by this point, still valuable, but the Italians were in the game at this point and had been for a long time. France IOTL even started their own in 1466, so whilst Silk might be good (it could be used to trade for West African gold, or other goods) - it isn't a great monopoly any more. Although it would certainly be an interesting trade good for England, and the only thing I've found on New World silk was in the 1640's. It could evolve to be of value - and I think the Mulberry trees might be able to survive the Canaries.
I think this would be a really fun timeline personally, especially if it is feasible for the Byzantines to assimilate the Guanches. Peaceful integration, despite being unlikely, would give them a starting population of 130,000 which would be quite a boost that could be constantly added to by importing indentured servants wishing to flee Turkish/Muslim lands (Or Hell, anywhere else if they were willing to work, convert, and learn greek)
As I mentioned above - there are more sources of people, but the Byzantines are going to have to learn to be tolerant - Gold, Silk and other trades are powerful lures to encourage immigration, but outright xenophobia and the out-of-the way nature of the Canaries would be a powerful disincentive until the Byzantines make the Canaries more important.