If necessity is the mother of invention, then serendipity is probably the father.
Many of our greatest inventions throughout history seem to have come into existence by chance, accident or sheer dumb luck. It just goes to show how easy it is to ignore a brilliant idea until it's staring you right in the face.
Everybody knows the now almost mythologised story of Alexander Flemming's discovery of Penicillin (which won't be showing up in this list, don't worry), but it turns out that he's not alone in making ground-breaking discoveries completely by accident. Although it would seem that the Flemming story might gloss over some of the more complicated aspects of that particular discovery, but never mind that now.
If history is to be believed, most scientists are just bumbling around their labs until a miracle occurs, like in the famous Sidney Harris cartoon. To be fair to the poor dears, when you're spending your time pushing the boundaries of scientific knowledge, it is very difficult to figure out what you're looking for until it appears.
In fact, it probably happens even more than we'd even know, but most scientists have got the brains to keep their mouths shut a pretend that it's what they intended to do all along.