I played Pitfall The Mayan Adventure back in the day.  It was fun to actually have a sling as a weapon in a videogame, even if just a platform game.
 
Well I'm a bit behind the times as far as videogames go, but I do have an old Gamecube.  I found last week that the Pitfall series continued past that initial game and includes a fun 3D game that works on a variety of platforms.  It's called Pitfall, The Lost Expedition.  I purchased a copy used on amazon.com to check out the slinging action.  Of course it took me a few hours before I was even granted my (semi-automatic?) weapon.  And a few more hours before I was given the ability to use it with maximum powered up force at a reduced rate of fire.  But it's a fun game, if more then a bit corny.  Scenic places to explore and lots of crazy monkey's and headhunters to knock out with sling stones.  
 
Just thought I'd mention it since video games in which the hero uses a sling as a weapon are quite rare.  
 

Hallo, fellow nerd-slingers  
 
I'd just like to show you the new game I'm playing - "Europa Universalis: Rome".  
No slingers there (just Velites, Princeps, Equites, and other funny wags), but... there's my town!  
 
It's the Capital of the Picenum Region  
 

 
The game is not that careful, though: I started a new game in 474 b.C., and Rome already got half Italy, while Regio V Picena was fully conquered only in the Third Century (and not even definitively, since in 91 it was the backbone of an important social revolution   ).
Greetings,
Mauro.