Either the edible or ornamental sweet potato vine can be started from harvested tubers provided the tuber is firm and healthy. Store the tuber in a cool (50 degrees), dry location until you want to start new plants. You can start new plants either by burying the potato in moist sand or suspending it in a glass of water.
I've also successfully potted up the tubers shortly after digging and grown them through the winter as houseplants. They require very bright and warm conditions to do this, however.