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. (PROSE: Shadowmind)

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

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. (PROSE: 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". (PROSE: 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". (PROSE: Shadowmind)

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

In the 10th century, the Fifth Doctor and Peri Brown visited Constantinople. (PROSE: Methuselah)

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. (TV: The Deadly Assassin)

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

Istanbul
By 1963, Constantinople had become Istanbul. Susan Foreman, while taking school on Earth, once forgot the time and called the city Istanbul. In consequence, her geography teacher made his entire 4G class chant, "It's Istanbul, not Constantinople!" (PROSE: Time and Relative)

In 2005, the Ninth Doctor fought Sontarans on a rooftop in Istanbul. Sally Sparrow saved his life from one of them, and gave him her Christmas homework for instructions for when the TARDIS would "burp" and jump forward in time twenty years without the Doctor. (PROSE: What I Did on My Christmas Holidays by Sally Sparrow)

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

According to the pseudo-narrative Torchwood website, 4th century BC Istanbul had sewers full of Weevils.