While living and traveling in Europe for 2 years, I found that many budget travels choose to stay overnight in the airport when they have an early flight. It helps save on hotel costs. I myself did it once, but not to save the costs. I spent a night at Düsseldorf International airport waiting for my connection.
If you want to do so, you should know that lying down on the chairs in the airport was not comfortable at all. On that night I was very tired and sleepy, so I slept tight for few hours. It was not a great night, but it was not bad.
And what I want to recommend you if you want to do so is you have to check whether the airport allows passengers to stay overnight. Not all the airport opens at night. One of my friend was about to stay overnight in the airport. She told me that she already took a bus to the airport and then she found out that the airport closed at night and passengers were not allowed to stay in the airport. Instead of saving money, she had to stay in an expensive hotel near the airport.
Besides, you should do some research about the airport that you are going to stay. Find reviews from people who stayed in the airport before. Not every airport welcomes passengers to stay overnight.
http://www.sleepinginairports.net/ is a great site that you can find several airport reviews. People share their stories and give useful information such as a quiet area in the airport. So, learn from other experience before deciding to stay overnight in the airport.
Last but not least, prepare a backup plan. Research the accommodations around the airport. In case that you are kicked out from the airport, you still have some choices.
Have you ever stayed overnight in the airport? How was your experience? Share your experience with me in the comment!