The sweet potato did without a doubt originate in the Americas; the best guess as to how it made it out into the Pacific is that Polynesians sailed to the Americas and traded or somehow acquired it their and then took it out into the Pacific where through long distance voyaging it was widely disseminated. The best evidence for this is the sweet potato itself; but as others have reviewed in other posts; there have also been chicken bones found in South America with Oceanic genetic markers- that is the domesticated chicken that went into the Pacific and that went to the Americas diverged a long time ago, but at some point in the last thousand years came back together in South America; doubtfully because a chicken managed to make it all by itself. Further plausible evidence is the coconut, which could not have floated against the currents and remained a viable seed to reach south america; Patrick Nunn has written and published about this last one, but as it was recent I haven't seen much else about it. The sweet potato and the chicken bones have both been addressed by well known archeologists and anthropologists like Patrick Kirch. Furthermore the name of the sweet potato kumara (or a variation of this) is a common name across Oceania for the sweet potato and appears in the Americas as a name as well.

Also a number of voyaging studies done by Geoffrey Irwin, who was just trying to demonstrate the strategies that could be used by Polynesians in order to find new islands to settle, which used computer models, proved quite accidentally that the Americas could be reached and returned from without a significant threat of loss to the crew. The success rate could be remarkably high if one knew the seasonal wind patterns that could carry you to the Americas and then switch and carry you back to Polynesia (or the Tuamotu/Society Islands most directly).

Further evidence has been in the form of Thor Heyerdahl who tried to prove a south american origin for the settlement of easter island and part of Polynesia, but who has ultimately contributed most to proving how easy it would be for polynesians to get home from the Americas if they arrived. Sir Peter Bucke an older maori scholar also theorized an American origin for Polynesians; imagining them migrating from South East Asia riding currents to the North West United States and then riding the currents from the Americas out into Polynesia (he was trying to prove Polynesians weren't related to Melanesians), and while his overall theory is more romantic than real; it demonstrated the clear possibility of riding currents and wind patterns across the Pacific.

Great evidence has also been the Polynesian voyaging society; Ben Finney, Nainoa Thompson, etc, who have established how accurately and precisely long distance voyaging can be accomplished if one is trained to look for signs in the environment and navigate based on stars at night. They have shown that one a place is known, sailing there is not terribly difficult.

Oral traditions don't have much to offer us; either because those that knew the tradition died during the terrible bouts of depopulation that struck much of Polynesia after the Europeans arrived; or because the oral histories are secreted away and not shared with outsiders- a quite distinct possibility. But either way; we don't have access to it.