Scrub potatoes thoroughly; halve any large ones. In a large saucepan, place potatoes and enough lightly salted water to cover. Bring to boiling; reduce heat. Simmer, covered, 10 to 15 minutes or just until tender; drain. Cool slightly.
In a large bowl, stir together mayonnaise, vinegar, mustard, sugar, pepper, and salt. Stir in the relish, red onion, celery, and dill weed. Add the potatoes and eggs; toss lightly to coat. Cover and chill 2 to 24 hours.
