Tremendously dumb ORD question
 

In the interest of using fewer award miles, I booked a UA round trip BWI-LAX itinerary that includes a connection at ORD. A six-hour layover, from just before midnight to right around 6. Obviously, no matter what, I'm checking the bag I'd otherwise carry, but once I land, then what? Is there anything in ORD that is still open then or am I better off just napping by the gate for a few hours instead?

You could head to the casino a couple miles away in Des Plaines (blue line train to Rosemont and then catch the shuttle bus):
http://www.riverscasino.com/desplaines/
Also:
http://www.sleepinginairports.net/usa/chicago_ohare.htm

If you're connecting on the weekends, the ORD Hilton is often really cheap and it's connected to the airport. I've seen it go for the AAA rate of $89. I'd spend that for four hours in a bed and a real shower. Some of the surrounding hotels with shuttles can be even cheaper (Hilton Rosemont: $71 for example).

I was just listening to a local radio jazz program and the announcer mentioned that a jazz act was playing that the Gaslight Club which is in the ORD Hilton. The website doesn't state hours but if I was on the type of connection the OP mentions, a couple drinks and some live jazz would be a happy timekiller.

Note, that once you exit you may not be able to get back before TSA checkpoints open at 4:30am.

Aside from the Rivers Casino mentioned above I would expect very few options around O'Hare overnight. There is a gentlemen's club within a short cab ride, if you're into that sort of entertainment. It's open until 4am.

You could also take a train all the way to the city and admire nighttime Chicago. A few places will be open until 2am (3am on weekends) so you can enjoy live music.

Actually, there are quite a few that are open to 4am on weekdays and 5am on weekends.

I'm still trying to figure out why this is a "tremendously dumb question." In my mind, "tremendously dumb" is asking, "What airline operates in the United Terminal?" or "Does ORD have restrooms?"