The name was actually probably in use before the conquest, as it's not Turkish. (This is a common misconception as most historians use Constantinople to refer to the city during the Byzantine period and Istanbul during the Ottoman, but in doing so it gives the inaccurate impression that there was an abrupt shift at the time of the Ottoman conquest.)

"Istanbul" was in relatively common use by the late 17th century, but the Turks did not rename the city until 1931; the official name قسطنطينيه‎ --Constantinople -- appeared on official documents throughout the Ottoman period.

The common belief is that Istanbul came from the medieval Greek "εἰς τὴν Πόλιν" ('is-tim-bolin), meaning "to the city" or "in the city" (in the same way that people who live around New York refer to it as "the city"); an alternative is that it's a corruption of "Constantinopoli" with the first and last syllabus dropped. Occasionally it is claimed that "Istanbul" comes from "Islambul," ("Full of Islam"), but that name doesn't appear to have been used much before the early 18th century.

source: Necdet Sakaoğlu (1993/94a): "İstanbul'un adları" ["The names of Istanbul"]. In: Dünden bugüne İstanbul ansiklopedisi, ed. Türkiye Kültür Bakanlığı, Istanbul.