Boil potato cubes until tender enough to easily pierce with a fork, but still firm – about 30 minutes. Strain, and place into a large non-metallic bowl.
Pour Italian dressing over warm potato cubes, gently stir to coat, cover and refrigerate overnight.
Several hours before serving, add green onion, celery, and radish slices. Fold into potatoes.
Add mayo (3/4 cup if you like a drier salad, more if you like it creamier), gently stir until potatoes and vegetables are evenly coated.
Gently fold in the egg slices, season with pepper to taste.