You know this one has to be important because it's part of the title of the book. Having the "heart of a samurai" doesn't mean that you have a super-bionic heart; it means you live according to the codes and creed of the samurai class.
For Manjiro, who is born into a much lower social class, this means weathering—in some cases, literally—the obstacles and difficulties in life:
Within him, Manjiro knew, beat a heart scoured by sand, pounded by waves, burned by sun, and polished by rain and wind. It would always be the simple heart of a fisherman, but perhaps it had also become the mighty heart of a samurai. (5.41.18)
Think of Manjiro's heart as a part describing the larger whole: His heart represents Manjiro himself, it is who he is at his core—strong, persistent, and readily compassionate for other people, even his greatest enemies (like Jolly and Tom). Manjiro really is "all heart," which is what allows him to be a samurai in both spirit and reality.