The two tiny metal balls that Captain Queeg constantly rolls around between his fingers are a symbol of his mental problems. The balls are like his security blanket, without which he would fall apart. The balls make the captain's nervousness and insecurity visible to the world. The turning point of the court martial comes when Greenwald badgers Queeg to the point that Queeg has to take out the balls to keep himself from falling apart.
Multitudes, Multitudes, Thomas Keefer's novel-in-progress about World War Two, is a symbol of the conflict Keefer has with the regular military. Whenever he feels oppressed by the conundrums of the Navy or by Queeg, Keefer turns to his writing. It is the last vestige of his former life, the jewel of his intellectual pursuits, and a constant source of debate between himself and Queeg, who thinks the novelist should be spending more time on his duties. When Willie finally gets to read part of the novel, he finds it entertaining, but utterly lacking in the literary merit that he expected from such an educated person. Because the novel is a failure, Keefer is a failure in what he may have thought was the only worthwhile thing he did aboard the Caine.
When Willie lends his hat to Harding so that the height-sick ensign does not have to vomit all over the ship, the hat becomes a symbol of their friendship and of Willie's acceptance aboard the Caine. It is also somewhat symbolic military stupidity. It was not necessary to make Willie and Harding climb to the crows nest, but they do, and the result of their submission to military ways hangs over their heads, waiting to wreak havoc.