This wasn't something that my property did, it was instead something we did at an old job; however, I always thought it would be a great idea for a Tenant Social... A Minute to Win It Party!
Have you ever seen the show? The games are extremely simple and use only household objects. Let's be honest, everyone loves a good competition, so hopefully that will draw the residents in. You could always offer a small incentive as well (if possible). Also, offering adult beverages to loosen the crowd up always helps (if you can swing it)! You could host it in a common area like lobby or game room, or better yet, at the pool or on a large lawn if the weather is nice enough.
I manage a 28-unit near-campus complex with lots of trees and green space. Two summers ago we organized four "On the Grass" outdoor movie nights. Each evening these started with an outdoor international potluck to tiki lamps. Tenants and friends of mine were invited. I was a lot of work but very gratifying. Far fewer tenants participated than expected. In fact, there were only about three each time. We plan on doing it again...just too much fun not to!
As of right now I plan to rent a banquet room in the bar/restaurant below the apartments. I am including some wine, beer and appetizers so far and a raffle. Just need some more ideas as far as themes, games, etc..