Here blinking is not about eyes. It is a mild, slightly comic way to show you are annoyed — a soft swear you can say in front of children, or in a sitcom. “This blinking printer” means you are fed up with the printer. It is British, a bit old-fashioned, and it usually sits in front of a noun like a grumpy little adjective. Americans do not really use it this way. You may also hear it as extra weight in a sentence: “I can't get the blinking thing to work.” Though actually, it has the same shape as the everyday verb blink (what your eyes do), but this annoyed use is a different job. I would not hang a big origin story on that. Treat it as a polite stand-in for stronger words, in the same mood as blasted or blooming.