Forget P vs NP and the halting problem! The real unsolved mysteries of computer science are the everyday nightmares we pretend don't exist.
That moving button that plays hard-to-get just as you're about to click it? Pure evil. And don't get me started on trying to send a simple file between devices—apparently easier than putting humans on Mars, yet somehow still impossible without sacrificing a mechanical keyboard to the tech gods.
My personal favorite: web developers somehow making simple text and images consume more memory than the entire Apollo mission. Because nothing says "modern web" like needing 16GB of RAM to read a recipe.