The Nazoraean Hypothesis on the Qur'anic Nasara
Today Christians consider the term somewhat derogatory, and the most common Modern Standard Arabic word for Christian is "musihi," which means "person of the Messiah" the same way "Iraqi" means "person of Iraq." However, at least for the first few centuries of Islam, "nasrani" was the only word in use, had no derogatory overtones we can detect, and was used by Christians and Muslims alike.
It has been a bit of a mystery, though, why that would become the Arabic term found in the Qur'an. Given the significance of Syriac as a spoken and liturgical language in the Middle East on the even of Islam, we might expect instead a version of the Syriac "Kristyan." While forms of nasrani/nasara do occur in other Middle Eastern languages, they mostly seem to follow on the Arabic usage chronologically. The exceptions are when it is put into the mouths of persecutors in accounts apparently designed to parallel an account of persecution in Acts 24.
In an article in the 2002 Bulletin of the School for Oriental and African Studies entitled "Nasrani and hanif: studies on the religious vocabulary of Christianity ans Islam," Francois de Blois has offered a plausible hypothesis. In late antiquity, there was a sect of Jewish Christians called the Nazoraeans who kept Jewish law and attended synagogue services, but also believed that Jesus was the Messiah. De Blois argues that the Christians of west central Arabia, the environment of the Qur'an, could have been of this group, surviving longer in that remote area than they did in regions closer to the centers of ecclesiastical power.
The lack of data makes this necessarily a tentative hypothesis, but de Blois notes it does address two other mysteries of the Qur'an's view of Christians. Several verses imply that Christians kept dietary laws, which Jewish Christians in fact did. Second, the Qur'an recounts God the Father, Jesus, and Mary as the Christianity Trinity, rather than God the Father, Jesus, and the Holy Spirit. Late Antique sources on the Nazoraeans show that at least some believed the Holy Spirit to be the same as the Virgin Mary, with the Trinity forming a sort of complete holy family which they venerated.