I think both public reactions and personal reactions determine this quality. Memorable characters appeal on a gamut of many different levels. Some are memorable because they are so good or loyal, some because they are a mixed bag of good and bad qualities, and some because they are just evil. Often the current social climate endears one character to that society. If the character fulfills the social needs for a certain type of person of any age, it creates a memorable character. The character can be strangely different or resemble a familiar person everyone seems to know.
Those publically well remembered usually come from mass media such as TV or movies, because the character comes alive in these media. I think sometimes this can depend and reflect more on the actor playing the part, bestowing the character’s personality and actions on the actor even if that isn’t what the actor’s personality embraces at all. Other memorable characters come from acclaimed books or lore.
These characters’ haunting presence may last only while a particular society needs that character's example, but sometimes something engrains the character in society for a much longer time. The ancient gods of mythology are memorable characters entrenched in society. So is Sherlock Holmes. Usually, the character shows traits the public admires, most often some form of extreme courage or perseverance. Sometimes they just exemplify what everyone would like to be like.
On a personal level, the character usually appeals to the mind’s psyche. Those characters speak on a personal level so are remembered. Because I like fantasy, science fiction, romance, history, mystery, and suspense, I remember fantastical characters, Darth Vader, Scarlet O’Hara, Angelique, . . . and the list goes on.
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