Place eggs in the bottom of a saucepan; add enough cool water to cover eggs.
Bring to a boil; cover, remove saucepan from heat, and let stand 15 minutes.
After 15 minutes, drain the water then fill the saucepan with cold water and ice. Allow to sit for 5 minutes.
Tap each egg firmly on the counter until cracks form all over the shell. Peel under cold running water.
Slice eggs in half lengthwise, and carefully remove yolks.
Mash yolks with mayonnaise. Add relish, mustard, salt, and pepper; stir well.
Spoon yolk mixture into egg whites.
Sprinkle with a bit of paprika.