Bibi is Bobinôt and Calixta's four-year-old son. He's a little smarty-pants. The first line of the story tells us that he knows a storm's coming (1.1) (although there's also an implication that the storm's arrival is so obvious that even a four-year-old can pick up on it). This is quickly followed by his observation that his mother, home alone, will be frightened. In this Bibi is more right than he can know – his mother is in some kind of danger or crisis point. While she first expresses this crisis as a fear of the storm, the real crisis comes in her encounter with Alcée and her decision to cheat on her husband. Bibi can't possibly know this is going to happen, but he is right to think that his mother will go through some kind of emotional difficulty at home, without her family, during the storm.
Like Alcée, Bibi shows wisdom in the eye of the storm, trusting calmly that everything is going to be all right: "Bibi laid his little hand on his father's knee and was not afraid" (1.5). It's Calixta and Bobinôt who are afraid at different points in the text. Bibi passes through unscathed, happy, and secure in the arms of his father – and in the return to his mother at the end.