One of those characters for me is Marnie Elmer from Winston Graham's book Marnie.
In his book, he describes a girl with so many issues:
1) Astraphobia - fear of thunderstorms
2) Erythrophobia- a fear of the color red
3) She's a girl desperate for her mother's approval
4) She's a kleptomaniac
5) And, she has intimacy issues
Why? Because something in her past is preventing her from dealing with life in a productive way. The book, like the Hitchcock film, looks at her life and why she acts the way she does.
We found out what the reason was behind it but her uniqueness makes her loved.
Who is a memorable character for you?