Place the eggs into a saucepan in a single layer and fill with water to cover the eggs by 1 inch. Cover the saucepan and bring the water to a boil over high heat. Once the water is boiling, remove from the heat and let the eggs stand in the hot water for 15 minutes. Pour out the hot water, then cool the eggs under cold running water in the sink. Peel once cold. Slice the cooled eggs in half lengthwise, and scoop out and reserve the yolks.
While the eggs are cooking, place the cut-up potato into a saucepan with water to cover, bring to a boil, reduce heat, and simmer until the potato pieces are tender, 10 to 15 minutes. Drain the potato, and let cool.
In a bowl, mash the reserved egg yolks with pickle relish, mustard, creamy dressing, salt, and pepper until well combined. Place the potato into a bowl, and coarsely mash with a fork. Lightly combine the potato with the yolk mixture. Stuff each egg half generously with potato salad, and sprinkle with paprika. Cover and chill until ready to serve, at least 20 minutes.