I was on an overnight layover on Lufthansa returning from Tel Aviv via Frankfurt to San Francisco. I debated whether to stay overnight at the airport. I went online and found a Hotel for $75 on Hotwire(It was the Mercure). For $13 I got breakfast. Not only did I have a comfortable sleep I swam in the hotel's indoor pool upon checkin, took a nice bath before heading to the airport and had a full breakfast.

I myself wouldn't sacrifice my health or sleep by sleeping at the airport but if I knew there was a quiet area then I may just sleep there. I've always wondered could you just sleep behind the ticket counter of your airline and be the first in line to check in?

One other option is to find an airline lounge that closes around midnight 1 AM and opens at 4 AM. In between these times ask the lounge agent where he or she suggest you sleep until they reopen.