This is a confusing name. These Lamanites have covenanted to be faithful to the God of Nephi and the God of Lehi and yet they choose a name which seems to imply that they are against father Lehi and Nephi. After their conversion, why would they take upon themselves such a name? Certainly, this group held no disdain for Nephi or Lehi.
The best explanation is that the group chose to be called after the righteous king Anti-Nephi-Lehi. He was the son of the king and a brother to Lamoni (Alma 24:3-6). Eventually he became king and the spiritual leader of this group. As we know, Book of Mormon peoples often called themselves by the name of their leader. Therefore, it would seem that the term Anti-Nephi-Lehi meant exactly what it seems to mean. When Lamoni's father had his other son, he decided to name him "Anti-Nephi-Lehi," a name which was in accordance with the entire Lamanite culture. When the people decided to call themselves Anti-Nephi-Lehies, it must have been because they liked the king, taking upon them his name without taking much thought for its original meaning. Later, when they join the Nephites, they are called "Ammonites" or "the people of Ammon" presumably because the Nephites didn't like the name, "Anti-Nephi-Lehi" (Alma 27:26).