Rub the turkey inside and out with salt, place in a pot big enough to hold it plus some, then cover it with cold water. Put a plate or something heavy on it to keep the turkey submerged, then cover the pot. Put it in the fridge at least overnight. From the book CookWise:
Water can flow in and out of cells, and normally it flows through cell walls toward the most strongly concentrated solution in the cells or tissues. When you rub salt on the surface of meat, the salt dissolves in the meat juices to produce a concentrated solution that draws moisture out of the meat cells. ...... However, if the water were not evaporating from the surface, the salt water on the surface would become diluted by the moisture. Soon the concentration outside the cell walls would be less concentrated than the liquid in the meat cell, so the salt water would flow into the cell. Meat cells contain a lot of water, but it is water that is bound and held by the proteins. Actually, the free liquid in meat cells is very concentrated with dissolved substances. This means that even concentrated solutions of salt water or salt and sugar water will be less concentrated than the liquid inside the meat cells. Braining is a way to increase the amount of liquid inside the meat cells - a way to make meat juicier.
More than you wanted to know?
Edit: Time is a huge factor with brining. Whatever you brine, make sure to keep it submerged in the salt water at least 3 hours, no less. Oh, and make sure it is completely thawed, if it was frozen, before brining.
Edit again: Just re-read your post. If all you do is stuff it with potatoes, yes, it will be too dry. So stuff it with potatoes and something to add moisture - could be a fat, a sauce, bacon, whatever.