According to the Myers-Briggs personality chart, every personality has another personality that is different from it in such a way that the two can't understand each other. This personality is your opposite in such a fundamental way that it seems impossible to understanding them. I am usually fairly perceptive. Generally I can understand peoples' motivations and sometimes even predict their actions and reactions. This is not an unusual attribute. It's called average social skills. It's also, I think, a very useful capability for writers.
But once in a very long while, someone comes along who makes no sense to me at all. In fact, only once has this happened in my not so long life time. Someone with whom I have conversations and feel that somehow I never quite understand what he meant. We speak and on some very basic level our thoughts don't line up and no communication occurs. Our little Venn-Diagram of communication bubbles, instead of overlapping, sort of bump each other and bounce off again. And then we stand there feeling confused—excuse me, I stand there feeling confused--I haven't the foggiest what he is feeling—and then I kind of shake my head and we go in opposite directions.
Then I stand around wondering what I did wrong, or said wrong, or didn't understand, and finally I decide that it can't possibly be me since I function perfectly well when talking to anyone else. And I can't escape the sort of haunting feeling that he is doing it on purpose. That he is watching himself play a part and watching me react to the part he is playing. And I hate not understanding. It's like an itch I can't scratch. A puzzle, or a riddle I want to know the answer to but someone took away before I could solve or memorize it.
It always occurs to me, shortly after any interaction with this person, that there is no way I could write him because I simply don't understand him. While there is nothing I can do about that (Except maybe tell him how confusing I find him and ask him to explain himself, though I don't think that would get me anywhere. He would probably say something opaque and leave me more confused than before.), I could certainly write about the severe befuddlement I am feeling. While I can't write my Enigma, I could definitely write one for my heroine. I understand her and so could make a person she doesn't understand. I think a good dose of confusion and irritation would benefit her immensely.
There is always a place for a riddle in a good story, whether it's a puzzle that guards a secret treasure or a person who makes no sense to the protagonist. Now that I mention it, a little befuddlement is probably good for me too. It might teach me charity, and zen-ness and communication skills. I think there is probably always a place for something one doesn't understand. Even if it is irritating.