The Turkish name of the Rose Mosque is Gül Camii. It is the former Church of St. Theodosia, with foundations from the late 10th or the 11th century, its brickwork is of the “recessed” type, typical for that period. It still preserves its original form. That is: a cross-domed church with side aisles supporting gaqlleries. Piers supporting the dome are disengaged from the walls, and the corners behind them form alcoves of two stories. The central dome and the pointed arches which support it are Turkish reconstructions, as are most of the windows. Outside many of the upper parts were altered in Turkish times. The minaret is from the classical period when the church was converted into a mosque.
I quote from Strolling through Istanbul, which considers one legend perhaps true, which explains its name: in 1453 on Theosodia’s feast-day, May 29, a large congregation appealed for the saint intercession. The church was decked with roses in celebration and when Turkish soldiers entered the city on that very day, the roses they found led to the name. Other legends may be found, but seem less convincing.
I visited the mosque with Raffaele D’Amato, who studies Byzantine churches, and took some pictures using a camera of mine, his pictures have a capital R in their name. I claim copyright anyhow.
I came across a site with reconstructions of Istanbul monuments around 1200, for instance the Gül Camii.