I am all for this! I think it could be similar to the "Chicks Dig This" icon that's already in place. You could have some serious fun with it, too. I'd label it something like "Cats Dig This?" or "Cats Dig This???"
My wife has taken a ton of snapshots over the years of our cat hanging out with me while I play certain games. The atmosphere has to be just right - he has to be in the same room, or not "busy" taking a nap, or something - but when it works, it works.
There was a moment about ten years ago when I was sitting on the floor of our then-apartment playing Killer 7 on the Gamecube. He jumped right up in my lap, settled down, faced the screen, and couldn't take his eyes off of it. His head would jump around whenever things got crazy (which, in that game, is often!) and he was so, so into it. When I was done playing, and had to get up to turn the system off, he didn't budge, and stared back at me and started meowing, as if to say, "No! Don't turn it off!" It was a trip!
By far, though, his favorite game is OutRun on the Sega Genesis. I've actually had him come running into the room when I started the game up, making noises like, "Hey! Wait for me!" He loves to sit in my lap and stare down the passing traffic, looking to the left or right whenever a car comes whizzing by. He looks over at the speakers as they pass by, and "follows" the car as it leaves the screen. Then, of course, he snaps his head back to the center, waiting for another car to come by. It's a riot, and best of all, he couldn't care less about the actual game, or how well I'm actually doing. He's a little less fond of me making up my own words to "Passing Breeze" and singing them to him while I drive, but hey, you can't have it all
There's plenty of videos out there of cats going crazy during Duck Hunt, pawing at the ducks as they scramble across the screen. My cat never seemed to care for that one (he positively *hates* the clicking sound of the Zapper) but if I were going to pick just one game as a bonafide "cat game," Duck Hunt would be it.