Place the turkey tails and onion in a 4-quart Dutch oven and fill halfway with water. Season the water with the salt and 1 teaspoon pepper. Bring the water to a boil, then cover and simmer over low heat until the meat falls off the bones, about 2 hours. Remove and discard the bones and skin; return the meat to the pot.
Add the potatoes and green beans to the Dutch oven and simmer until the green beans are very tender and the potatoes can be pierced easily with a knife, about 20 minutes.
Meanwhile, melt the butter in a small skillet, then add the garlic and remaining teaspoon black pepper and cook for 2 minutes. When the beans and potatoes are done, scoop with a slotted spoon to a serving dish and drizzle with the garlic butter.