I think the first decision will be whether it is worth it to you to go through the hassle of leaving the terminal (and having to do the security screening again the next day).
If you leave the airport you won’t be very far from the ocean. You could put your stuff in a locker and take a walk to the shore. There is a nice oil refinery on the waterfront not far from there.
Another option might be to try and find a convention to crash at one of the hotels nearby. Swipe one of the nametags off the table and then see how long you can enjoy free drinks and finger-food until they realize your not with the convention.
If you’re an art-lover, there is also the Getty Center. The hours might not work out too well for you (the connecting flight leaves at 12:30 right?) but the architecture at that place alone is worth the trip, in my opinion. You could astound the locals by taking the bus! (The bus driver would probably be very happy for the company, being acustomed driving the route for weeks between passengers.)
Of course, if you stay in the terminal then I think the best plan is probably to find a computer connection and just spend the night online chatting with forumosans.