First of all, I find that it's almost impossible to write any sort of character without some simple things. I like to know what the person looks like, what their voice sounds like, and a vauge idea of what their surroundings would tend to be. In other words, Jane Generic here is a beautiful woman who favors 1950's style boiled wool suits and pillbox hats. She speaks extremely politely in a whisper. She works for the front desk at an office building and the main character wonders about her history.
Why are these helpful?
Well, all of these help the author find a role for the person. Here, you could say Jane Generic fits into the Girl Friday (or Christie's Miss Lemon) archetype of the efficent secretary who's completely collected. The role helps fill in your voice (does she have a polished formal tone, or is she incongrously accented and informal?) It also helps you have a setting (her office) and her appearence (typical office attire? something else?)
Now, once you have that, you can start to fill out a matrix. This is a way to know how someone reacts without, necessarily understanding why they'd act that way.
Let's say you've got the archetype of "Strong Silent Bodyguard."
"Strong": When are they strong? What situations would you expect them to show their strength? Not show their strength?
"Silent": Does their silence become a problem? When do they talk? When don't they talk? How do they use their silence?
"Bodyguard": Who do they protect? Why does this person think they're being protected? Do you think they're right? If there was a similar person, would they be protected? How does this protection work?
These questions serve to map out someone on a scale of how they fit within the broader scheme of things. You could say, for example, that Jane Generic is a terrible fighter, a good debater, secretive with her private life, and has a mysterious box that she keeps locked in her desk. This doesn't tell you why she is that way, or even how she came to be the woman that she is, but you do know how she'd act around some situations.
Obviously, this is heavily dependent on what will happen with Jane. If Jane'll never see a fight, it doesn't matter how good she'd do in one. If she'll be in a lot of debates, you'd do better to examine how likely she is to participate, and then look into things like debating style, common platforms, behavior around various debators, etc.
Another way to look at things is consistancy. Many people react consistantly to certain things, and they also tend to have certain exceptions. A list of constants and exceptions is a pretty simple way to make someone with some quirky character traits, but a seemingly well filled out personality.
So, in conclusion, a mysterious character can be easier to write if you can place them on a scale of various issues or if you can have a list of constant behavior and exceptions. Once you have that, you merely check the situation at hand to your list, and you're good to go.