The Red Sox cap is a symbol of that thing which helped Trisha to keep calm, to stay balanced, even to escape from the oppressive reality in some way. Trisha takes it on and feels more confident after. Even when she is at the hospital, her last wish before falling asleep is that her father puts this cap on her. Of course this thing wasn’t actually a true helper of her, this support was just in the girl’s mind. But she believed in this symbol, so it helped her and made her strong and brave.