MBA 13" 2011 (with TB) as a display for PS3
You'll need to look for HDMI capture boxes. The best one for your situation would seem to be these Blackmagic products, specifically the Thunderbolt one (it says "shipping in January").

I doubt there'll be many (if any) HDMI-USB capture boxes, except maybe USB 3.0, which your Air doesn't have.

Of course, if what you want to record (and maybe watch) is HDCP protected, you won't be able to. Be aware that any digitizing solution will introduce lag so this is not at all an optimal solution for gaming where you might care about reaction speeds.