Hagios Giorgios Church (Saint George/Aya Yorgi) is located on one of two summits of Büyükada. After reaching the central point of the island (known as Birlik Meydanı Square), you should climb a steep cobblestoned path on foot to get there (it's too steep for a fayton, as well as a bike, to climb).
It takes about 35-40 minutes. Although the church building itself is unexceptional with nothing really fascinating, the backyard of the church offers some very beautiful sights of the other islands and the sea.
On April 23rd every year, which is considered as St George's holy day, a crowd of seemingly tens of thousands attend the church to make wishes. Wishmaking rituals that day range from usual burning a candle to climbing the cobbled path on bare feet to untying wool balls all along the path.
The final part of the path that day is as crowded as a rock concert (except very early in the morning, like 6AM, it's reported), because police officers let people in in groups of 10-15 at once, to avoid an overcrowding inside the church. If you decide to burn a candle that day keep in mind that nearer the church you are, cheaper the candles being sold around (for example, in 2007 it was YTL 2.00 near the quay, while the going price very near the church was YTL 0.50 per candle).
But better of all would be to buy the candle inside the church for a donation (it's up to you how much to pay) as none of the profit of the candles sold on the streets benefits the church.
And a sidenote: Most of the people attending the church and waiting for a blessing from the priest upon exit that day are non-Christian Turks, but there is nothing surprising about that: This is Turkey , where east meets west (and vice versa) and cultures truly mix.
On the other summit of Büyükada lies the abandoned Greek Orphanage (Rum Yetimhanesi). Originally built as a hotel in late 19th century, this completely-wooden, 4-story building is the second largest wooden construction in whole world (the largest in Europe).
It's dangerous to enter the building itself (because it's slowly decaying), and also forbidden - but you may try to coax the guard for a permission at the garden gate.
Both the eastern and western side of Büyükada is full of wooden Victorian-style mansions dating back to late 19th/early 20th century, similars of which have been bulldozed in the rest of Istanbul (with the exception of neighborhoods on Bosphorus banks) to make way for concrete, multy-story apartment buildings.
The ones on the western side (right side when looking out of quay) seem more splendid. Just don't be surprised and don't start looking for them as soon as you get off the ship: Around the quay is more like a town centre. They are located about 15 min walk away from the quay.