Here in Hawaii the Hokulea is a big deal (http://www.hokulea.com) because they have taken modern day people, and using the Hawaiian methods of navigation they can still get around without modern technology and figure out where to go. Showing it wasn't just Polynesians sending out random boats hoping they would hit an island.

The origin of Polynesian sweet potatoes being from an old visit to South America is the most plausible origin story, but the lack of anthropological evidence is also a powerful dissuading factor. There's no oral stories of either the Polynesians or the South Americans of such a voyage taking place, like there was for the Viking colonization of Vinland (recalled in the Icelandic sagas).

Note, however, that the Australian aboriginals' oral histories (songlines) lack any reference to coming from elsewhere, although DNA clearly demonstrates otherwise.

In such cases, if any given generation omits an element from its oral history (by choice or mistake), it is lost forever.

There are two orders of magnitude difference in time though. Australian Aboriginals have been on the continent for 40-60 thousand years, Polynesians made it to Hawaii 800-2000 years ago.