Unlike my bug-eating, backpacking brethren, I am known as an anthropologist who likes her creature comforts. Nonetheless, even the most softened anthropologist has had to occasionally spend the night in an airport. I found a fantastically funny and informative website that ranks the various airports for their sleepfulness, gives tips on where to catch some zzzzs and useful hints about what and what not to do in the process.
My own personal favorite, Singapore Changi, is listed there at #3. They have a mini-hotel where you can shower and sleep for the few hours between a layover. And you can literally spend half a day wandering the shopping arcades or catch a bus to town and enjoy the "Night Safari" at the zoo. My second favorite is Amsterdam's Schipol. The best thing is its proximity to the city. This made in simple to dash in, explore a couple of museums, and dash back all in time for your departing flight.
No Japanese airport makes the top ten. It's odd that we -- the creators of the capsule hotel, the "unit bath," and the Washlet -- cannot design an airport that you'd want to spend a few hours -- or build an airport with easy and inexpensive transport access to the city. Although to be fair, I just read a news clipping in Asahi Shimbun yesterday that the new Nagoya International Airport will have an onsen bath with a view of the runway. No mention was made whether departing passengers will be able to catch a final glimpse of naked Japan at its best.