An interesting question, and one that I don't have an answer to right off the bat:
Sometimes I find eccentric characters quirky and fun, other times I find them too unbelievable and annoying. What are some of the more outrageous characters you’ve read, and how do you feel about them?
I would think Agatha McGee, from Jon Hassler's work, would strike some as an "eccentric" character. I love her, and she has come to be part of our "family language." Agatha is a retired school teacher and sees many new things as the coming of the Dark Ages. Since I often agree with her, my sister and I call each other, saying, "I'm having an Agatha Moment!" and then go on to relate some thing that makes us shake our heads.
I'll try to think of other eccentric characters and update later. I suppose I'd say weirdness for the sake of weirdness or shock irritates me. As a part of a fully developed character, I like it.
A favorite of mine is Mrs. Ariadne Oliver, in some of the later Poirot novels by Agatha Christie. I have a sort of theory that she's a bit of a rebuke to Dorothy Sayers, who made the Lord's own fool of herself by first falling in love with a fictional character, and then writing herself into her own novels and having Lord Peter fall helplessly in love with her. Very embarrassing. Mrs. Oliver is a slightly comic turn, who is forever going on about a woman's intuition never leading her astray. She's always wrong, but wrong in a way that points to the truth, and she's a really enjoyable character.