Many have wondered about the name of Anti-Nephi-Lehi. Elaborate explanations have been given to explain how it doesn't mean what it seems to mean. Yet, for a moment, let's entertain the possibility that the name really meant "against Nephi and Lehi." The king of the Lamanites had at least two sons, one was named Lamoni and the other was named Anti-Nephi-Lehi (Alma 24:3). Given Lamanite traditions, it would not be surprising if he gave one of his sons a name which was derogatory to the Nephites. Later, of course, all three of these Lamanite kings were converted to the gospel. Anti-Nephi-Lehi was made king prior to his father's death (Alma 24:3). He became a great king and spiritual leader. Therefore, it is entirely possible that the people took the name of Anti-Nephi-Lehi not because they were against the Nephites, as the name suggests, but because they were taking upon themselves the name of their leader, a common practice in the Book of Mormon.
Furthermore, it is not surprising that the Nephites preferred to call them by a different name, "the people of Ammon" or "Ammonites." Apparently, they were never known as the Anti-Nephi-Lehies after they began to dwell with the Nephites because the term Ammonites was considerably less threatening.