Cover potatoes with cold salted water in large pot. Boil; cook 12-15 minutes until fork-tender, not mushy.
Drain; spread on baking sheet to cool about 10 minutes while preparing dressing.
Whisk mayonnaise, mustard, relish, vinegar, sugar, salt, pepper, and celery seed in large bowl until smooth and combined.
Gently stir warm potatoes into dressing to coat and absorb it.
Fold in celery, onion, and eggs until evenly distributed.
Cover; refrigerate at least 4 hours or overnight to meld flavors.
Stir gently before serving; adjust salt and pepper if needed.