Place eggs in a large pot and cover with cold salted water. Bring salt water to a boil and immediately remove pot from heat. Cover pot and let eggs stand in hot water for 15 minutes. Pour hot water out of pot and run cold water over eggs until completely cool. Peel eggs and cut in half lengthwise.
Separate the yolks from the whites of each egg. Mash the yolks in a bowl. Add mayonnaise, pickle relish, sugar, yellow mustard, vinegar, salt, and pepper to egg yolks and mix until smooth. Spoon yolk mixture into the egg whites and sprinkle paprika over yolk filling. Refrigerate until chilled, at least 1 hour.