This is what my "crazy" sister made for my sister-in-law who is laying over in Seoul, Korea for 10 hours. My sister-in-law decided to go with the option 2 and she thoroughly enjoyed lay over in South Korea!
Two Options Recommended
1) If you are tired, stay in the airport. You can still have fun.
2) If you want to go out and experience both local and tourist areas + local shopping, take the option 2.
Option 1: If you are tired, stay in the airport
1. Spa on Air
Since you are tired from the flight, you can first rest at Spa on Air (B1, East side) It costs 15000 won (about $15) and extra 10,000 won ($10) for private sleeping room.