Quirkiness is a quality of being slightly eccentric or unconventional. If your best friends love your blue hair, taste for mayonnaise sandwiches, and habit of singing instead of speaking, they appreciate your quirkiness.
While quirkiness is usually a little strange or peculiar, it's also somehow fascinating. The quirkiness of your brother's sense of humor might leave you shaking your head as you laugh; and the quirkiness of your dad's baking probably results in startling flavors, some of them delicious (and some not so much). This noun comes from quirk, which some etymologists believe was once a term for an unexpected flourish or twist in weaving.