As we don't do your homework for you here, if you've read the book, the answer part should be easy!  But if I got to be Oprah, I would ask:

* These rumors of your husband's infidelity during his long journey away from you must have been hard for you to endure.  How has it affected your marriage?  Do you believe that he's been faithful?

* Upon your husband's return, he tested your fidelity, despite rumors of his own infidelity.  Was that a difficult obstacle to overcome?

* How has your marriage changed since he's returned?  Have you seeked marriage counseling or religious counseling?

* How did you come up with the ideas you came up with to keep your suitors at bay?

If I had the opportunity to interview Penelope, here are some of the questions I would ask:

• How have you struggled in raising your son all of these years? What has been difficult about not having a present father?
• Why have you allowed the suitors so many privileges at your home?
• How has it felt to be so overpowered by this great number of suitors?
• What is your relationship like with Anticlea, Odysseus' mother?
• How is your relationship with Telemachus? Does he hold any ill-will for his father since he has been absent all these years?
• How does it feel to be a queen without a king? Do you have any great responsibilities in the land that you've struggled through?

These are questions I legitimately have of her when I read The Odyssey. Asking about her relationships should draw great emotion.

If I were to interview Penelope, my questions would focus on how she endured the absence of Odysseus, maintaining her fidelity and her responsibility as a mother, and how she felt when he returned.  I think that questions like, "Did you ever doubt that your husband was gone forever," would bring out some very emotional and zealous responses.  I would also focus on her raising Telemachus on her own and how this played into her conception of self.  Finally, bringing out the return of her husband would also get her to speak about her fidelity and her perception of the suitors that wanted to win her hand.