As her speech competition approaches, Stargirl-turned-Susan talks an awful lot about that silver plate. It's the trophy for the champion, and our girl wants it, bad. She plans on bringing it home to show off to her adoring fans that she knows will be waiting to greet her at the school. In a way, she sees the silver plate as her only hope for being accepted again. In that sense, it's also her only hope for making Leo happy.
When she does win that plate, she holds onto it for dear life. The only time she doesn't hold it is when she dances with Leo at the nightclub at the hotel. When they arrive home the next day and it is clear that there is no mob waiting to cheer her victory, she lets that plate drop to the ground by the car, and it's the clatter heard 'round the world. With that plate fall all of Stargirl's hopes of fitting in. Instead of clinging to it, as a symbol of her self worth, she lets it drop, because she no longer cares. Sure enough, the next day, she's back to being Stargirl, and doesn't care that she sticks out like a sore thumb. In fact, she relishes it.
After she drops the plate, her father picks it up and gives it to Leo, "with a strange smile." What do you make of that? Maybe Leo, who tried hardest of all to make Stargirl fit in, deserves to have the symbol of her own inability to fit in. Maybe he's meant to learn that you just can't make an extraordinary girl ordinary.