Matt, with one day only it would be madness to leave Paris. Go to Versailles on another visit. You really need a full day to see & appreciate Versailles. By the way, don't limit your hotel choices just to be near the Tour Eiffel. I have often stayed in the 15th arrond. and the bus I take back to my hotel passes right by the Tour so I see it every day.
If you want to be right by Notre Dame there is a charming hotel on the top floor of the historic Hopital Hotel Dieu which is just steps away from Notre Dame. It used to be just for the families of patients but is now available even unto thee and me.
A good website for information on the metro system (never called subway in France) is parisbytrain.com