What is the connection of the word "gift" to the book Hatchet?

The hatchet that Brian uses in the story was a gift from his mother.

When Brian’s mother gave him the hatchet, neither of them knew how important it would be. She just assumed that he might have some adventures in the wilderness. She never thought the plane would crash.

Brian was very emotional when he said good-bye to his mother. His parents were getting a divorce, and that was all he could think about. His reaction to the hatchet is ambivalence. He does not want to tell her that he doesn’t really want it, and that he is upset.

And he would normally have said no, would normally have said no that it looked too hokey to have a hatchet on your belt. Those were the normal things he would say. But her voice was thin, had a sound like something thin that would break if you touched it, and he felt bad for not speaking to her. (Ch. 1) 

Brian lets her put the hatchet on his belt. He does not give it much thought. When he gets into the plane, it is still there. When he notices it, he is a little embarrassed. The pilot says nothing. The plane is small, and Brian and the pilot are the only ones onboard.

His mother gave him money for food, in case he was “stranded” at the airport. Brian gets stranded, but not at the airport. The pilot has a heart attack, and the plane crashes into a lake in the middle of the wild.

And on his belt, somehow still there, the hatchet his mother had given him. He had forgotten it and now reached around and took it out and put it in the grass. There was a touch of rust already forming on the cutting edge of the blade and he rubbed it off with his thumb. (Ch. 5)

The hatchet is the only survival tool Brian has. He has a little trouble finding food, but is able to use the hatchet to create a fishing spear, and later to cut into the plane and get the survival kit. The hatchet is the reason Brian lives, so it was quite an appropriate gift.