Examine your genre and allow it to influence the choices your character must face. For instance, crime stories naturally lend themselves to exploring issues of justice and injustice: At what point do revenge and justice converge? What does that require of this character? When is preemptive justice really injustice?
Love, romance and relationship stories often deal with themes of faithfulness and betrayal: When is it better to hide the truth than to share it? How far can you shade the truth before it becomes a lie? When do you tell someone a secret that would hurt him? For example, your protagonist, a young bride-to-be, has a one-night stand. She feels terrible because she loves her fiancé, but should she tell him what happened and shatter him—and perhaps lose him—or keep the truth hidden?
Fantasy, myth and science fiction are good venues for exploring issues of consciousness, humanity and morality: How self-aware does something need to be (an animal, a computer, an unborn baby) before it should be afforded the same rights as fully developed humans? At what point does destroying an AI computer become murder? Do we really have free will or are our choices determined by our genetic makeup and environmental cues?