Look, Billy loves dogs. Loves them. Seriously, this kid gets sick because he wants a dog so much. So, obviously they're important. It's hard to say if the kid Billy can see past their cute little paws and soft ears, but the adult Billy—the one writing this story—sees them as symbolizing the best parts of the human spirit: loyalty, courage, and determination.
Buddie is the first dog we meet in a story about the relationship between man and dog, so he's got to be important, right? Sure he is. This is the dog that is the catalyst for the whole story we are about to hear.
What's really weird is how much Billy seems to know about this dog:
To him it made no difference how long the road, or how rough or rocky. His old red feet would keep jogging along mile after mile. There would be no crying or giving up. […] Through the rains, the snows, or the desert heat, he would jog along, never looking back. (1.31)
Billy is clearly projecting his experience with Old Dan and Little Ann onto this Dog. If you think about it, there are actually several similarities:
From this little list, we can pick up a few things: (1) dogs help Billy be the best person he can be—kind, caring, and brave; (2) they're closely connected with poverty and hard work and doing things the DIY way; and (3) they represent something about the nature of childhood.
That's a lot of meaning to pile on the back of one scrappy species.
Even for dogs, Old Dan and Little Ann are something else. Their weird, mystical connection lets them do things that no ordinary hounds should be able to—track the ghost coon; help kill a mountain lion; stay out all night in a snowstorm to tree a raccoon. So what makes them so special?
Billy's parents believe that the dogs answer the prayers of both Billy and the family by providing the money they need. Sort of like a two-for-one special: Billy gets his dogs, and the family gets the money it needs.
They tell Billy that God only answers prayers if you say them with all your heart. This certainly makes the dogs seem mystical in nature, like angels or spirits. And Billy learns to have faith in God through having faith in his dogs. Even in almost hopeless situations, Billy refuses to give up his belief in his dogs. At the very end, when he sees the red fern, he believes in them again—only this time, belief in them means belief in God.