I overnight a lot at the airport in Seattle, and sometimes Portland. Should work in most other big cities too.*
Great place if you can blend in, don't have too much stuff (my 50L backpack and/or shoulder bag is all I carry), and can be out of there around 5am (tip: put a post-it note on your bag or clothes that says "Wake me up at 5am" people will wake you up then.) Several airports also offer free wifi (I know Seattle and Portland do), which is a nice bonus.
I've only been bothered once (out of 50-60 nights) by security as to whether I was flying in or out. Usually if you say you just flew in from another big city (my go-to is Houston, I'm usually wearing a Texans ballcap) and you're just waiting for public transit to start running, they'll be ok with it.
*I've heard the northeast airports are very unfriendly to people spending the night there, but it's just hearsay.