I usually travel SA, I've spent a lot of nights stuck in airports, ORD, SFO, BOS, and DEN etc. etc..
I just sleep on the chairs usually. Spend the night on the phone, walk around looking at all the shops, I enjoy strolling around the concourses for exercise, or looking out at the airfield. ORD is a cool place to hang out, the tunnel in the United concourse is fun to walk around a couple times.
Edit:
Also traveling from an SA point of view, and having a greater understanding on how airlines work help me feel comfortable at airports over night. Even if I miss one because of cancellation, I don't care. Usually I get stuck from flights being full, which can't be helped since I used to pay nothing to $20.
Its funny, I wrote an essay about the different kinds of travelers once, usually there's the family travelers who don't understand airlines and flip out at the first news of flight cancellation, flight full, delays etc.
Next there are business employees/travelers they're generally pretty good.
Finally the SA travelers, who don't care how/when they get there, just that they do!
I split it up into more, but generally that's how I see it.