Fun arts neighborhood downtown. Sit down and enjoy a Pisco sour and a large platter of meat and watch the passersby and the art.
Stay nearby. I used to recommend the Sheraton for high service but I haven't been down that way in awhile and the service has apparently gone downhill. The Singular is nearby and very nice but quite expensive. I'd either plan on heading back to the airport or grabbing a cheap hotel nearby - there's plenty in the city district, and that's what TripAdvisor is for.
Most of the national museums and whatnot are nearby. You could easily spend a day going to one or two of those plus dinner.
There's an old Pinochet prison camp, Villa Grimaldi, that is quite moving but a long cab ride and you'll need your Spanish (though there is an English audio guide I think). It's more authentic than the purpose-built Museum of Memory and Human Rights but the museum's easier to get to and more accessible.
Check into the hotel and ditch the bags, head into town, then come back to sleep at an airport hotel. Way easier than either dragging the bags back to the airport at the crack of dawn or paying for both a hotel room and left luggage. Also makes the transport cheaper as it's easy to take the airport bus without the luggage.
I had an overnight layover there on the way home from a trip to Easter Island. I was surprised by how nice a hotel it was.
Just down the street from the Estadio Olympico
Anyone who travels internationally frequently or has been to the Santiago airport in particular, I would like to know if it would be more advisable to pay to leave my luggage at the airport for 24 hours (if that's possible there) or take it to the hotel with me and bring it back to the airport in the morning.