In Hatchet, Brian's ability to keep going even when times are tough is really put to the test. Throughout the book, hope is often the only thing that keeps Brian moving forward. In the early part of the book, of course, Brian is hoping and expecting to be rescued at any time, and his main focus is on keeping himself alive until that happens; so keepin' on keepin' on seems pretty natural. But once the rescue plane passes him by and he's forced to give up that hope, Brian hits rock bottom. When he finally breaks out of his depression, we see a new Brian—one who is far more self-reliant, and full of what he calls "tough hope"—the ability to persevere, and to continue to plan for the future no matter what.
The only thing more important than the hatchet to Brian's survival is his ability to persevere.
Brian isn't very good at persevering in the early part of the book—he's really just doing his best to survive until he's rescued. Only after he's given up hope of being rescued does he really start to develop perseverance.