For example you could show slightly different versions of the show to different viewers, of you could subtly alter the show when viewed on second or third viewing. Or the show could change depending on the time of year that you watch it. This kind of breaking of the fourth wall and breaking of fiction from reality could fit extremely well into a Black Mirror type of story. I'd love to hear what others suggest.
As far as interactivity, and 'choose-your-own-adventure' style of story telling I think that will probably get boring pretty quick. We already have this kind of story telling with games such as Heavy Rain and Beyond: Two Souls and while many people view those games as innovative forms of cinematic games, I would say they are pretty tedious and non-immersive forms of cinema.