I think you can if you are airside, but they do close the lobbies and baggage claim areas overnight.

You have plenty of time to try to find a reasonably priced hotel for an overnight stay though. If it were me, that's what I would do. And then you'd have a decent nights sleep and a decent meal. The airport food options are not great except perhaps for Stinger Rays in the interisland terminal. If you do end up staying you can go walk over to the interisland terminal in the morning and have a nice breakfast there. I think I had banana pancakes last time I arrived early from Asia and had a couple of hours to wait for my i/i flight home to Maui. Watch out for the prices! You can tell them you are flying to Asia to skip the Ag check when you walk back to the overseas terminal.

-David