When Brian's mother gives him the hatchet, she explains that he will be able to use it on his camping trip with his father. Brian doesn't seem to like the gift very much, and he thinks that "it look(s) too hokey to have a hatchet on your belt." However, Brian senses that his mother is upset ("her voice was thin, had a sound like something thin that would break if you touched it"), so he doesn't voice his dislike of the gift, instead quietly attaching it to his belt for her to see. This quietly restrained, sympathetic reaction says a lot about Brian, who at this point is angry with his mother for cheating on his father.
Brian's mother's reaction to seeing the hatchet attached to Brian's belt ("Just like a scout. My little scout") suggests that she also bought it for him because she likes to mother him. She likes that she can still take care of him, in this instance by providing him with a tool that might be useful to him on his camping trip.
Brian's reaction to the hatchet later in the story is very different. He understands that it is vital to his survival. Indeed, in chapter 8, Brian thinks to himself that "The hatchet was the key to it all" and that "The hatchet was the answer." Indeed, Brian uses the hatchet to construct a shelter, build a fire, and fend off wild animals.