Imagine there was maize - a plant with definitive American origins - in Europe before 1492, and Europeans, despite their own linguistic differences, were all calling it "hominy" (which is the English approximation of word for maize among the Powhatans). That'd be pretty compelling evidence that somehow, at some point in time, Europeans and Powhatans met and exchanged maize before 1492.

This is essentially the situation for Polynesian-American contact. Sweet potatoes are a common and widespread crop in Polynesia from New Zealand to Easter Island. However, the plant originates in South America. Maybe the plant somehow got swept out to sea, but this seems unlikely as its not tolerant of saltwater. Additionally the techniques used to propagate the plant are the same as those employed in South America. Perhaps that's just a coincidence though. Two unconnected cultures might both develop the same techniques for farming the same plant just because the plant is well suited for those methods.

What's harder to explain without direct contact is the name. Throughout Polynesia, the plant is known generically as kumara (the various cultivars have their own unique names, too). In Quechua, it's known as kumar.

The Rapanui of Easter Island likely reached South America sometime before they lost their forests and their ability to construct long-distance voyaging craft. They were aware of the small island between themselves and South America, known today as Isla Sala y Gomez, but to Rapanui it's Hotu Motiro Hiva - the Islet on the Way to Hiva. To the Rapanui, "Hiva" is the term for the distant lands, which suggests they were also aware that beyond Hotu Motiro Hiva there was yet more land, which could only be South America.

The sweet potato line of evidence is currently the most compelling for the Polynesian contact in South America. Suggestive chicken bones have also been discovered in South America (chickens are one of the major domesticated animals among the Polynesians and the only one the Rapanui maintained at the time of European contact). Initial tests indicated that these bones were from Polynesian rather than European chickens and were from Pre-Columbian times. Later tests have muddled the issue a bit, calling the dating into question and revealing that the genetic affinities is not so clear cut. Another line of evidence are a set of Polynesians skeletons found on Mocha Island, immediatetly off the coast of Chile. However, the skeletons have not yet been dated, and may be from the 1800s when many Rapanui and other Polynesians came to South America, voluntarily or by force.