Well, earlier records can attest that it was derived from ancient eastern/Zoroastrianism symbols. Though later the eastern parts of the Roman empire used them too. I think it's most likely that the Turks borrowed from the Byzantines, as they seemed to have modeled themselves after them in many cases, though that's a little debatable as a lot of Muslim flags were seen with the symbols during the crusades. I think there was a specific Turkish meaning with them, but I'm not so acquainted with them as much as I'd like to be.

While the crescent by itself dates back far earlier in the Islamic world, the star and crescent as it is seen today is probably Turkish in origin. The Turks (the ruling ethnicity of the Ottoman Empire) were originally a central Asian people that later migrated south into Anatolia. Prior to their adoption of Islam, their 'pagan' belief system held the sky to be sacred, and celestial iconography was common among Turkic peoples.

The white-on-red star and crescent as seen on the Turkish flag today did not come into use in the Ottoman Empire until the late 18th century.