There are several small examples of courageous acts throughout the movie, such as the time when Joe and Simon Birch peer into the bag brought to them by Ben (which contains a stuffed armadillo), when he steps up to the plate to finally "swing away," or even the scene of the Christmas play, where Simon works up the courage to touch the girl playing Mary.
But probably the two scenes that show the most noble form of courage, are right at the very end. Simon Birch and Joe are accompanying a bus full of third graders on a church field trip when the bus loses control on some ice and careens into a nearby river. As it slowly fills up, there is more than enough time to get the kids out safely, but the bus driver panics and flees immediately. As the bus is in pandemonium, Simon Birch has the wherewithal to command everyone's attention, and then lead them out and to safety. One boy gets caught by the shoe and Simon holds his breath to get the boy's foot free, then has to swim out of the opening of a small bus window to his own safety.
The very next scene is another notable scene for bravery. Simon is laying in a hospital bed, dying, and saying goodbye to Joe. As Joe holds back tears, Simon stays completely calm. He has a sense of peace about him that might normally be reserved for someone elderly, who is ready to go. He bequeathes Joe his baseball cards and ends with, "I gotta go now," never once losing control of his emotions nor his sense of peaceful resolve. Most would consider this courage in the face of death remarkable for anyone, but especially for a 12 year old boy.