There are multiple mini-games that fall into the entirity of the International Airport Olympics. Just lkie the olympics has multiple events, so does this.

Baggage Claim Psychic
This game is points based. You and your date mosey on down to the baggage claim and post up in an inconspicuous spot. Wait around for a flight to get in and the bags to start rolling down the conveyor belt.
Your goal: match luggage to travelers.
Each time your correct (which is not that often), you get a point. These tend to end a lot like soccer games (low scores, ties). But oh man.. when you get one right... it's time to gloat.

Then, once the losses are too much to bear (because again, unless you're a real psychic, matching luggage to travelers is NOT an easy task), it's time to move over to a security checkpoint and play "Bigger, Better, Backstory"

Bigger, Better, Backstory
This game is all about crafting a story about a person's life who you presumably know nothing about. The idea is that you give a potential detail about the intended target, and then your date "Yes, And's" you and gives another detail.. but just a little more fantastical. Each additional detail should get more and more incredible until you move onto the next person. Then, you can continue to interweave and bring back in previous "characters" and stitch together back-stories for the entire airport population.

The airport doesn't close. They're open 24 hours a day (well, except some of the businesses inside--they close).
You can really choose any time you want, but here are a few tips that may come in handy:

• When are a bunch of flights coming in?
  You can look up arrivals/departures on the airports website iirc. You don't have to get this granular, but you can check out to see if more flights seem to get in around one time or another
• Plan on grabbing a drink or bite to eat?
  If so, then you should probably rule out certain times when those restaurants will be closed. Also, consider what is OUTSIDE of the security checkpoint. Unless you have a ticket or a way past TSA, you are pretty much relegated to the food/drink options in the general area of the airport. Don't fret though, there are going to still be at least a few places to choose form.
• Where are you going to park?
  It'd be a good idea to get an idea of the parking situation prior to going. How much is short term parking? Do they take credit cards or cash only? Are you carpooling or meeting each other there?
  All pretty simple things, but it'll make you much more comfortable and relaxed when you already have an idea of how things are going to work--it's why performances have dress rehersals.

Granted, if it's a small or private airport, this doesn't really work. The people watching and the food options will be nil, so best to stick to the large airport (especially the international) variety.
If you aren't sure where the closest airport is to you... I wouldn't even suggest Google'ing it--I'd tell you to just not do this.
You must live hours from an airport, so this will become anything but a cheap date if you were to follow thru. haha