Place beet in a small saucepan with water to cover; bring to a simmer and cook until tender, about 15 minutes. Drain and cool. Grate beet on medium holes of a grater. Place grated beet in a bowl; stir in mayonnaise, mustard, and relish.
Place eggs in a saucepan; cover with water. Bring to a boil, remove from heat, and let eggs stand in hot water for 15 minutes. Remove eggs from hot water, cool under cold running water, and peel.
Cut each egg in half lengthwise; place egg yolks in a bowl. Mash egg yolks with a fork; stir beet mixture into egg yolks. Season with paprika, salt, and pepper.
Place egg whites cut-side up on a serving platter. Spoon yolk mixture into egg white halves; top with cilantro.