The gun is the catalyst that reveals the true nature of his relationship with these characters. His acquisition of the gun reveals that:
1. He is willing to lie to his mother and father in order to achieve some sort of independence (which to him, the gun represents). So, he is feeling undervalued, and needs to prove his worth.
2. His mother will give into him; he is able to manipulate her, using his father as the manipulating tool.
3. He is afraid of Mr. Hawkins (he lied about the donkey), but underneath it all resents him and his authority and power over him and his family (probably one reason he skips town at the end instead of fulfilling his promise to Hawkins).
4. He considers himself better than the other workers (they were sharecroppers) on the farm; he refuses to work off his debt, instead taking his gun and seeking a "better" life. He seeks the gun in the first place as a way to prove his worth, and set himself apart from the other workers on the farm. It is a way to gain power in a very powerless lifestyle.
The gun helps to highlight the cracks in these relationships, and, ironically, just how immature Dave really is. Looking at it this way, the gun is actually quite an important "character" in this short story.