There used to be Stack Exchange Swag and a Stack Exchange store but that's gone by the wayside in recent years.
Really the most effective promotion for this site is Google and link-dropping in other forums. (Obviously we prefer appropriate link-dropping like "Oh they talked about this over on the Aviation Stack Exchange, the answers there might be interesting." over spammy "Hay ewe guise! Plz2Go here! KTHXBAI!" link-dropping.)
The site is also a good resource to give to student pilots or aspiring aviators around the airport as you noted, but we don't really have any premade promotional materials for that purpose. You could certainly create your own though (a simple tear-off flyer can be banged out in Word, I think they even have templates for it now).
If you decide to create your own flyer make sure to observe the Stack Exchange trademark guidance (which is really pretty common-sense stuff - you'd be hard-pressed to screw it up), you can post it here and we can ping the community team to get an eye cast over it if you want to be extra-sure.
(I bet others would probably want to make use of any promotional material you create too).