The most cost effective way would be to ask for a very late checkout, followed by a dinner then get a good book to read while waiting at the airport.

You should also ask the hotel if they will accomodate an "incidental stay" (which is a stay for less than a full night) on your last day if you want to hang around longer than a late checkout would allow. As another poster mentioned, this will cost much less than a full night's stay.

You should note that you will get MR points credit for the dollars spent but you will not get a night's credit towards elite status for incidental stays.