I think I figured out why.
Most nights, we throw the animals out of the second bedroom/computer room on our way to bed. (Some nights we don't because they're not there, but that's fairly unusual.) In order to do this, we usually have to wake the cats up.
So what's happening is Zilli is being awakened abruptly in the middle of the night. He staggers around for a couple of minutes trying to wake up, then realizes he's got to go to the bathroom. By this time we're in bed, so we get to listen to him scrape around in the kitty litter. Thus relieved, he's all awake and cheerful, and goes forth to see what's going on. Only it's the middle of the night and the humans have gone to bed, so there's *nothing* going on, and so he runs around the entire house saying, "HEY! Where ARE you guys? HEY! I'm AWAKE! HEY! Let's PLAY! HEY! Listen up! HEY! Come out and play! HEY!"
I am *convinced* that's what's going on. :)