Can I preserve fresh potato gnocchi by drying it?
Surprisingly, the answer seems to be a qualified yes, however the texture is not the same. The dried gnocchi turned out to be much better if fried after boiling; see below. 

Here are the results of my experiment:

I dried a small amount of the fresh Gnocchi by placing them in a 150 degree (Farenheit) oven for about an hour, then turning off the oven and leaving them in all day. It is very dry here, so that did the trick. 

I boiled the dried gnocchi after several days of storage. They took about ten times longer to boil than their fresh counterparts. Even after they were fully cooked, they were not as soft as the fresh gnocchi, and they never regained their original size or shape. Overall, they were denser, firmer, and less sticky. Not desirable traits necessarily, but they were quite edible.

I decided to pan-fry a small batch in about a tablespoon of oil, and these turned out wonderfully! I have pan-fried gnocchi before, and it tends to be hard to keep them intact and keep them from sticking. The previously dried gnocchi, however, did not stick and held up to vigorous frying. The result were gnocchi with a crispy exterior and soft (if somewhat dense) interior. The crispy outside and flavor reminded me of samosas. These weren't just edible, they were great. I plan to dry about half of my next batch and use the dried gnocchi exclusively for frying.