In a large Dutch oven, bring potatoes, salt, and water to cover by 1 inch to a boil over medium-high heat. Cook until tender, 10 to 15 minutes. Drain.
In a large bowl, stir together mayonnaise, relish, green onion, vinegar, mustard, and pepper until combined. Gently stir in eggs and potatoes. Cover and refrigerate for at least 1 hour. Garnish with green onion and paprika, if desired.