Place potato, carrots, and eggs in a pot and pour in enough water to cover; bring to a boil. Reduce heat and simmer for 15 minutes. Remove carrots using a slotted spoon, cool on a work surface, and dice. Continue simmering eggs and potato for 5 more minutes. Remove eggs with a slotted spoon and transfer to a bowl of ice water until cooled, about 5 minutes. Peel eggs and dice. Continue cooking potato until tender, about 5 more minutes. Drain pot, cool potato on a work surface; dice.
Combine potato, carrots, eggs, peas, pickles, onion, dill, salt, and pepper in a bowl. Stir mayonnaise into salad until evenly coated.