Dr. Robert Dean Luginbill, a Christian Professor of Classics outlined a plausible theory for the origin of the word Christian.
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:
- 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").
- 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).
- 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]".
- 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.
For manuscript evidence, see an excellent discussion and list of manuscript evidence courtesy of @curiousdannii showing the transition from χρηστιανος (with ETA) to χριστιανος (with IOTA). Dr. Luginbill's article cited above also commented on how the scribe of the extant Codex Sinaiticus meant to write χρηστιανος:
As to the ancient manuscripts, they all have the word in all three places and their testimony is identical – with one critical exception. The best and earliest codex of all, Sinaiticus (aka Aleph, א, the ms. is now on-line; see the link: Sinaiticus), has instead of Christianoi (Xristianoi/), Chrestianoi (Xrhstianoi/) – and it has this reading in all three places where the word occurs. Therefore it is impossible, in spite of the Nestle-Aland tentative suggestion for Acts 26:28, for it to be an itacism (i.e., a popular misspelling based on third/fourth century shifts in pronunciation, something of which this manuscript is, it is true, replete). For one thing, I find no parallel for changing a long "i" (iota) to a longe "e" (eta) in this manuscript (and the unusual spelling would not have happened three times by mistake). Equally interesting is the fact that in all three cases, the right vertical stroke and the horizontal stroke of the ETA have been erased to produce an IOTA (yielding the traditional spelling). This is very unusual. Sinaiticus was corrected many times, and each generation of correctors had their own discernible "tics". But simple erasure without further comment seems to be unprecedented. Moreover, the empty space left by the erasure is, in all three cases, not filled up. This shows that without any question the scribe of Sinaiticus deliberately meant to write "Chrestian" in all three instances; it was not a mistake. The plot thickens when we consider that two of the earliest secular references to Christianity, Tacitus, Annales 15.4, where Tacitus talks about the Christians being persecuted by Nero as "Chrestians", and Suetonius, Claudius 25, referring to Claudius' expulsion of the Jews mentions a certain "Chrestus" as responsible, we find precisely the spellings one would predict if these authors (or their sources) were deriving their information from the same tradition which the spelling of Sinaiticus suggests.