The significance of the title has to do with the word "wait," and what it means, in this context, to wait for something. Waiting is something you do until something else happens. Clearly, that "something else" is different for the father and the boy.
The boy, superficially, is waiting to die: he doesn't understand the difference between Celsius and Fahrenheit, and thinks his temperature of 102 will be fatal. What's left to our imagination is what is going on in the boy's head, if he thinks he is going to die. He seems listless and distracted; he tells his father he doesn't have to stay if he doesn't want to; in hind sight, it's clear he is being incredibly brave.
It's not so clear whether the boy thinks the father knows he will die, however. If we think he does, then the boy must see his death as a kind of terrible secret not to be openly spoken about. In that sense, his emotional detachment is a kind of masculine denial of emotion, something he has learned from his father.
The father, on the other hand, has more mundane things to wait for. He's literally waiting to dispense medicine. He knows the boy is in no real danger, so he is also waiting for him to improve, or at least cheer up a bit. His reading of the pirate story is less about entertaining the boy than getting him to hurry up.
At one point, he does leave to go shooting, and Hemingway renders the details of this excursion—how many birds were killed, how the dog was, slipping on ice and dropping the gun—perhaps even more vividly than the details of the actual story about the boy. One way to think about that is that, for the father, the business of hunting and the business of nursing his son are all the same. Hemingway of course makes no comment on any of this; he's good at making dots in that way and leaving it to the reader to connect them.
This concept of "waiting," then, both suggests that the present moment of the story is less interesting somehow than what will come after and highlights the emotional separation of boy and father, who, on the one hand, are too busy "waiting" for various things to emotionally connect, and, on the other, are nonetheless bound to each other through this mutual denial of connection.