Preheat oven to 350°F. Soak dried tomatoes in very hot water to cover in a small bowl until softened, for about 10 minutes. Drain and cut into thin strips.
Heat oil in a small nonstick ovenproof skillet (about 8-inch diameter) over medium heat. Add onion; sauté until softened, about 5 minutes. Add potato and ham; cook another minute or two. Add tomatoes.
Meanwhile, lightly beat together the eggs and cheese in a small bowl. Add to ingredients in skillet. Cook for 3 minutes longer.
Bake in preheated 350°F oven for 10 minutes or until eggs are set. Serve immediately.