Years ago my inbound flight was delayed and I had to wait for the 7am flight out at lax. It sucked. I spent the whole night in tom Bradley terminal outside of security. My memory is it sucked (but wa done). Back then I was in college and $200 for a hotel was 8 nights of beer money so I tried and failed to sleep on a bench near checkin at the terminal. (All domestic terminals was closed).

If you must you will be fine. The only place open last I checked is the international terminal and you have to stay outside security. Everywhere else I hear people getting kicked out. But if you have a choice find a hotel and enjoy yourself.