In Heart of a Samurai, Manjiro totally want to be a samurai. The only trouble is, he is "a fisherman's son" (1.2.33), and at this point in Japan, there's not a whole lot of upward mobility. In fact, Manjiro's friends even laugh at him when he announces that he "hope[s] to become a samurai" (1.1.30). Becoming a samurai is just beyond the reach of a poor fisherman's son.
So Manjiro can never be a samurai… or can he? What does being a samurai, in Manjiro's head, actually entail? Manjiro wants to "be like the noble samurai of old times: heroic warriors who were loyal to their lords, and who studied calligraphy and poetry as well as the art of fighting" (1.2.37). Which—if you think about—is something he kind of accomplishes while he's in America since he receives an education, especially in the liberal arts.
And when he returns to Japan, because he's been through and has learned so much, and because times are a-changin', Manjiro's actually made into a samurai. It's totally unprecedented, and totally awesome.
Manjiro is made into a samurai partially because the Edo court needs his knowledge of America to negotiate with the Americans, but the larger reason is Manjiro's ability to survive with ingenuity and perseverance: "Within him, Manjiro knew, beat a heart scoured by sand, pounded by waves, burned by sun, and polished by rain and wind" (5.41.18). That's a pretty strong heart. It's one that's not only "the simple heart of a fisherman, but perhaps… also… the mighty heart of a samurai" (5.41.18). Hey there, title.
The "heart of a samurai" is just another way of saying that remaining honest to his core traits—openness, inquisitiveness, adaptability—is what allows Manjiro to achieve his dreams. He's always been a samurai inside, but in the end he gets recognized for these qualities. A nice, heart-warming message all wrapped up in the title of the book.