What is the significance of the cup of coffee Billy has when he camps at Bluebird Creek with his father and grandfather in Where the Red Fern Grows?

The significance of Billy's drinking coffee is that it makes him feel like a man, not a boy.
This is the first time that Billy's Papa and Grandpa have given him this special privilege. Previously, it was only the grown-ups who'd drunk coffee. At home, Billy's never allowed to drink the stuff. But out on the trail, where men are men, it's a different story. Billy has earned the respect of Papa and Grandpa, so it seems only right and proper that they should let him drink coffee just like a man.
But although Billy may feel like a man, he still displays occasional evidence of immaturity, such as when he hears the eerie hooting of a couple of screech owls in the distance. Billy takes this as a bad omen, but Papa and Grandpa tease him for being so superstitious. It seems that there's still a long way to go, and much more coffee to be drunk, before Billy finally makes the transition to manhood.