In the book Hatchet, what character traits and qualities does Brian use to face the challenges of hunger, loneliness, and lack of shelter?

While Brian is initially presented as just an average boy, he does indeed possess a number of character traits which help him survive in the wilderness. Foremost is his resilience. Things do not go well for him with his initial adjustment to life in the wild. He eats berries that make him sick, gets attacked by a skunk and later a porcupine, and injures himself numerous times. However, despite realizing that he can't rely on a rescue from the outside, he makes the very conscious decision to not give up hope and to make every attempt to persevere in the face of overwhelming challenges. At times, he is very close to giving in, but in the end, his resilient nature comes through and gives him the will to keep on surviving. It is this resilience more than anything else which gives him the fortitude to carry on.
Furthermore, Brian is very observant. He takes the time to notice all the things around him and consider how they can help him survive. For instance, by paying close attention to tracks in the sand, he is able to locate a cache of turtle eggs (chapter 10).
Eventually, Brian also learns to stay calm in the face of danger. Panic has gotten him to trouble before, such as getting attacked by a porcupine or eating bad berries. After some time surviving in the wild, Brian learns that he will fare better if he keeps a level head about him when he senses danger. We see this come to play in chapter 13 when he sees a wolf. Instead of reacting out of fear or panic, Brian recognizes that the animal poses no harm to him if he remains calm.