Yes, I dig my ornamental sweet potatoes after the first frost, and store them
in my garage in a big container of dry vermiculite. Dry peat moss or sand should also work fine. In the spring I will put them in small containers outside in potting soil and wait until they grow sprouts, and replant them. I sometimes cut up the tuber before replanting. I found that reusing the same tuber for a few years leads to pitted, sickly looking tubers and smaller plants. You can also easily root sweet potato vine in water and replant a start that way.

After digging the potatoes, let them air dry for several days. Store them as Todd suggested above. In spring when they sprout slips, you can let the slips grow until you get a couple of pinkie-finger sized leaves. Use a grapefruit spoon and get a piece of the potato with the slip. Pot them up and stand back as they grow.

Oh BOY, yes, you can save those ornamentals for next year. My chartreuse vines were quite prolific last year, covering everything in sight, and even trying to cross the street. Where they touched the ground, they put out new roots. They even climbed up my rose bushes. After the vines withered, I discovered some huge tubers, one as big and round as a grapefruit, with purple flesh. I did not try to eat one. I left my taters in the ground, but of course Mobile had a fairly mild winter. I was not sure they would regrow anyway, but indeed they did. This year they did the same thing, and my neighbors pulled them out of my roses while we were gone. They love heat, and they love water. I planted them thinking they'd shade the soil and keep the other plants' roots cooler. But they grew like KUDZU almost. I even planted some for a neighbor, to cover an old stump in her yard. This winter, I will keep some of the vines indoors like a philodendron and see what happens. :)
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Thanks everyone for the responses. I'll probably have plenty of potatoes to dig this fall, so I'll give it a try.