As we noted above in "Billy Budd, Christ Figure," the narrator's characterization of Billy is jam-packed with the narrator's own ideas about who Billy is and what he stands for. The phrase "Noble Savage" does not actually appear in Billy Budd. It is linked with the romantic and naturalistic thought of the philosopher Jean-Jacques Rousseau, who imagined that society has contaminated man, and that the ideal men were those who were more intimately connected with nature, those who were, as they said in Rousseau's time, "savage." Rousseau's phrase might capture the worldview of the narrator: Billy Budd is his example of the noble savage.
One of the ways that the narrator characterizes Billy is as a simple man. As in, Billy is really simple. What's weird, though, is that the narrator doesn't mean to be putting Billy Budd down by emphasizing how unintelligent he is. In fact, the narrator worships Billy for his simplicity.
Consider some of the early descriptions of Billy: "He possessed that kind and degree of intelligence going along with the unconventional rectitude of a sound human creature, one to whom not yet has been proffered the questionable apple of knowledge" (2.10). A bit later, the narrator says, "Of self-consciousness he seemed to have little or none, or about as much as we may reasonably impute to a dog of Saint Bernard's breed" (2.11). As degrading as such descriptions are, the narrator's purpose is actually to compare Billy to Adam before the "Fall of Man" in Genesis. In other words, he compares Billy to uncivilized man.
As the narrator explains it, there seems to be something more dignified and pure about men before they became civilized. It is as if "savages" were closer to human nature because they were not yet forced to take on all of the different roles that society thrusts upon them. As the narrator says when speaking of virtues: "they will upon scrutiny seem not to be derived from custom or convention, but rather to be out of keeping with these, as if indeed exceptionally transmitted from a period prior to Cain's city and citified man" (2.13). The narrator sees virtue not as originating from "custom or convention," but as somehow preceding it. What one might term the good came before civilization and is thus separate from it.
For the narrator, Billy's mental simplicity is akin to complete sincerity. He thinks that Billy's good-heartedness means that he is lacking a certain mental fold that allows for lies and evil intentions to slip into the human mind. The result is that the narrator treats Billy as a romanticized version of…dun dun dun, the Noble Savage!
Now, we poke fun at this view just because we encourage you to get outside of it. Notice how often we point out that this view is attributable to the narrator. Just because the narrator portrays events in a Rousseau-type light doesn't mean you have to. That said, to get outside of this viewpoint you'll have to use the text itself, to find points where the narrator undermines himself without even knowing it. For an idea of how to do this, check out our argument in "Writing Style."