Some characteristic or habit that's odd or distinctive is a peculiarity. Your brother's peculiarities might include always wearing a red knit hat, even in summer.
A peculiarity is a quirk or strange feature. You might be self conscious about the fact that your ears are uneven, something you think of as an obvious peculiarity. Another kind of peculiarity is a quality that is specific to someone or something: "One peculiarity of people from the Midwest is that they call soda pop." This "special characteristic" meaning is the original one, which makes sense given the Latin root, peculiaris, "of one's own."