Büyükada has two monasteries but any links with a Byzantine history seem to be unknown or non-existent. There are supposed to be some bits of a monastery to St Irene remaining in the area of the island called Maden, the most substantial remains being those of the cistern. I haven’t found these yet but here is a picture from the 1950s.
[1]
Büyükada, Monastery oif St Irene (from Erdenen, Orhan: İstanbul Adaları)
[2]