What is Odysseus' secret?
This question is difficult to answer because it is not clear what secret is being referred to here. Odysseus keeps many secrets throughout the course of Homer's Odyssey, but his primary secret is his true identity.
Throughout the epic, Odysseus constantly tries to conceal his identity. One of Odysseus' most signficant mistakes comes when he reveals his name to the Cyclops, who then prays to his father Poseidon so that the sea god will cause great trouble for Odysseus.
One of the keys to recognizing Odysseus is a scar that he has on his leg. In Odyssey 19, Odysseus' aged maidservant recognizes Odysseus by this scar and in Odyssey 24, Odysseus uses this same scar to prove his identity to his father Laertes.
Another of the keys to revealing Odysseus' identity is his bow. At the end of Odyssey 21, when the disguised Odysseus strings this bow, his disguise vanishes and he reveals his true identity.
Perhaps this question refers to Odysseus' marriage bed. In Odyssey 23, Penelope tests Odysseus by claiming that she had their bed moved. Odysseus immediately becomes very upset because he knows that a "great secret went into its making." Once Odysseus tells Penelope "the true secret of our marriage bed", she recognizes him as her true husband.