"A Day's Wait" is an interesting Hemingway story in that it focuses on the actions and thoughts of a child, something that most Hemingway stories do not do. Also, unlike most Hemingway stories, it has a bit of a surprise ending that changes the story's thematic focus.
Because we find out at the end of the story that the sick little boy mistakenly thought he was about to die, it really takes a second reading to appreciate the story fully. Only then do we realize that the boy is trying to deal with what he believes is a death-sentence: a fever of a hundred and two.
The boy is most certainly being brave. Look at how hard he tries to remain calm, believing that he will die soon. Most importantly, look at how he thinks of others, even though he believes his death is imminent:
At the house they said that the boy had refused to let anyone come into the room.
“You can’t come in,” he said. “You mustn’t get what I have.”
Here, the boy thinks that he is protecting the health of others. For a child to think about someone else’s welfare in the face of his own death is definitely courageous.