This never made sense to me. "Anti" means against or contrary to. Why would repentance be signalled by identifying yourselves as being against the early heroes in the Book of Mormon who are clearly portrayed as pious and righteous? Most apologetics for this revolve around claiming that "anti" in this passage doesn't mean "against." For example, this incredibly lame article at scripture central claims that "anti" is really an untranslated Egyptian word that means "one of." Why Joseph Smith would mysteriously confuse people by leaving this word untranslated boggles the mind and makes zero sense.And it came to pass that they called their names Anti-Nephi-Lehies; and they were called by this name and were no more called Lamanites.
However, maybe this can this be explained by Mosiah Priority. Being anti-Nephi-Lehi makes no sense with what comes before Mosiah in the current Book of Mormon, but might it have made sense with the contents of the 116 pages? I realize this calls for speculation, after all we really don't know what was there. However, we do have clues that Joseph himself was very skittish about recreating the history that filled the 116 pages. Jarom, Omni, and Words of Mormon are just long lists of kings with absolutely no content. There seems to be a lot of time and history for the names of Lehi and Nephi to have taken on more negative connotations. Perhaps there were other characters who were evil named Lehi and Nephi. We know that Joseph Smith wasn't opposed to re-using the name Nephi as he did so in Helaman. Or perhaps political or military movements rose up associated with the names of Lehi and Nephi that tainted the names by association. In any case, it seems like reading "anti" as being just what it usually means may be more natural than I had previously thought.