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 "Chrestian""Chrēstian" (meaningχρηστιανος, 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.
- 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.