Istanbul

Istanbul was the name used by at least the later half of the 20th century for an Earth city which had previously been known as Constantinople, and before that, Byzantium. (NA: Shadowmind)

Byzantium
Byzantium was, according to Benny, the oldest name of Istanbul. (NA: Shadowmind)

In the 4th century BC, the city's sewers had a Weevil population, according to the Torchwood Institute. (WEB: torchwood.org.uk)

Byzantium remained the name of the city around the time of Jesus Christ. The First Doctor, Barbara, Vicki and Ian visited the city in 64 AD, and found it to be a city struggling with three groups vying for power: the Jewish Pharisees, the Roman Empire and the adherents of the fledgling Christian religion. (PDA: Byzantium)

Byzantium begat New Byzantium. The Seventh Doctor once told Ace that he thought New Byzantium stole the name because the two cities had a similar geography, both cities being set "on an isthmus of land, cut through by a strait linking two seas". (NA: Shadowmind)

Constantinople
The name switched to Constantinople in the early part of the first millennium. Benny once told Ace that the name switch was in honour of Emperor Constantine. She went on to say that Constantinople was "the capital of the Eastern Roman Empire until the 15th century, when it was captured by the Turks". (NA: Shadowmind)

In the 3rd century, the Doctor encountered Fenric in Constantinople, where he defeated him at chess and trapped him in a flask. (DW: The Curse of Fenric)

At some point, the Doctor bought a hookah in Constantinople, which he would later use as part of a ruse to evade the Chancellery Guard on Gallifrey. (DW: The Deadly Assassin)

In 1914, the Seventh Doctor and Ace visited Constantinople to locate and dispose of a malfunctioning Krivani interstellar drive. (ST: Ante Bellum)

Behind the scenes
Narratives in the Doctor Who universe have not, as of 2011, particularly defined when the city's name changed.