Terry is Manjiro's best friend (besides the Captain) in America. A guy his age who doesn't have an issue with Manjiro's race, Terry's a prize—he's also the complete opposite of Goemon, who is all about caution and negativity.
Think of Terry as an engine that doesn't stop. In fact, when he goes to a photographer to get his daguerreotype done, the photographer keeps telling him to stop moving and to shut up. He warns Terry that the daguerreotype will turn out blurry because of Terry's constant movement. And indeed, that's exactly what happens: "the photographer had been right—Terry was a blur, as if already on his way" (4.34.29). So it goes with this guy.
But where is Terry going? To California, of course, for gold. So besides being a good friend, Terry's other important function in the book is to get Manjiro off his butt and out to California with him. As he points out to Manjiro:
"You're a grown man, John. You must be… what? Twenty-two years old? In our country, young men strike out on their own. The captain and his missus wouldn't expect anything else." (4.34.22)
Terry's your (stereo)typical, young American male, all ready to go and make his way in the world. He's a natural driver of the book's plot, and a natural friend to Manjiro.