In a small pot of boiling water, boil the potatoes until tender. Drain. When cool, slice thinly.
Cut the hardcooked eggs into quarters and then cut each quarter into half.
Lay out the lettuce leaves on a serving platter or on individual dinner plates. Layer the ingredients on the leaves, diving the potato slices, egg pieces, chunks of tuna, black olives, tomatoes, cucumber slices, and peppers. Drizzle the wraps with the dressing and sprinkle with the basil. Serve the lettuce wraps with additional dressing on the side.
If making ahead of time, do not dress until ready to serve.