Dr. Robert Dean Luginbill, a Christian Professor of Classics outlined a plausible theory for the origin of the word "Christian" with extra-Biblical and manuscript support.
As many suspected (the OP, guest37, Ray Butterworth, and some commentaries I consulted), Dr. Luginbill also believed that it's the outsider who started the name "Chrēstian" (χρηστιανος, meaning "the goody-goody bunch" or "members of the household of Goody-Goody"), while the believers preferred to designate themselves "Followers of the Way" or "Disciples".
But a few decades later since the name stuck, the believers changed it to "Christian" (χριστιανος) and started to use it for self-designation, to mean followers of Christ, which obviously work well until today.
Here's a quote from Dr. Luginbill's answer for the 4 stages he proposed:
1) The unbelieving gentile inhabitants at Antioch took notice of this new group which was trying to convince them to join (evangelism recorded in this chapter). Not having a Jewish background, the word "Christos" meant nothing to them, so that referring to these people in terms of "the anointed One" would have been a bit too much for them when referencing this group whose activities and beliefs (to the extent they understood them or were interested in understanding them at all) seemed ridiculous. Ready at hand, however, was an excellent pun on the name that seemed to fit them to a tee. Rather than "followers of the anointed One", they were "the goody-goody bunch" or "members of the household of Goody-Goody" (Greek chrestos, xrhsto/j, often meaning "good" or "moral" in Hellenistic Greek). Besides being a good pun, this appellation caught precisely the sanctified behavior that characterizes believers and which contemporary unbelievers in particular found so odd (cf. 1Pet.4:4: "They think it strange that you do not plunge with them into the same flood of dissipation, and they heap abuse on you").
2) This pun was so good, in fact, it caught on rapidly as the main way of identifying this new sect. Believers preferred to be called "brethren" or "disciples" after scriptural parallels, and also occasionally as "followers of the Way" (cf. Acts 24:14). As the new appellation caught on, it was apparently always used with a large measure of scorn, and believers were certainly aware of this. For that reason, they did not apply this term to themselves, except when making a point of the ostracism to which the secular world often subjected them (this accounts for both the rare use of the term in the NT and the fact that in all three instances it has this negative connotation of external abuse).
3) Eventually, Christians reacted, and began to say, in effect, "You've got that wrong! We are followers of Christ! Not of somebody named 'Goody-Goody'!" When this began to occur, the correction "Christian" for "Chrestian" most likely became a self-designation. We see the transition no doubt already accomplished by the time we get to Tertullian, writing at the turn of the next century. He actually vents his spleen against the mis-pronunciation "Chrestianus" (which may actually have been the original form changed by Christians to reflect the Lord's rightful title): "But Christian, so far as the meaning of the word is concerned, is derived from anointing. Yes, and even when it is wrongly pronounced by you "Chrestianus" (for you do not even know accurately the name you hate), it comes from sweetness and benignity" (Apology 3.5). This same attitude is to be found in Ignatius' letter to the Romans (3.2): "[I pray] that I may not only be called a 'Christian' (i.e., by others as a term of abuse), but be found as one [in truth]".
4) Later readers and correctors of the earliest manuscripts began to see the spelling Chrestianos as an error for Christianos. Oral readers in the scriptoria where manuscripts were copied sometimes consciously sometimes unconsciously made the change in pronunciation as they read resulting in the later editions showing no trace of the original ETA (the scribes never knew the difference). What happened in the case of Sinaiticus is similar. I doubt that it was a conscious effort to obliterate and cover up any trace of the earlier, correct reading. Rather it most likely struck the person who made the changes (at what point, early or very late, we cannot tell), as a mistake that needed correcting. He probably thought he was doing us all a great favor. Fortunately, the evidence has not been entirely erased, even if two thirds of the ETA has in all three passages.
Dr. Luginbill concluded:
What does all this mean? Are we Christians or Chrestians? While the word "Christian" might never have been coined except as a corrective, defensive mechanism, we certainly have the right to use this fine designation to describe ourselves, whether or not it originally came from the Bible. For we are in very truth, "members of the household of Christ", and proud of it.
In the Name of the Anointed One, our dear Lord and Savior Jesus Christ.