I have only heard of the evidence from the dispersal of and the Polynesian names for the Sweet Potato, both connecting it to the South Andeas
I think the first thing to keep in mind is that we have quite little evidence of Polynesians in America.
Perhaps more surprising is that we have pretty clear evidence of Amerindians in Polynesia - this is a surprise to many researchers, as the assumption for years had been that Polynesians would have reached South America. They key evidence of this is genetic study made last year, that showed that Polynesians on several islands carried DNA that could be traced to a specific people on the Pacific coast of Colombia.
Why is this significant? Because it indicates the contact was seemingly, perhaps surprisingly, made from South America to Polynesia, and not the other way around. It fits with a relatively small population of COlombian natives reaching some islands in Central Polynesia, such as the Marquesas Islands, around 1200 AD, about the time Polynesia was settled, and while contact was still maintained between the islands. Therefore, after the initial admixture, the descendents spread to other islands, so that the Native American DNA is present in many Polynesians.
Conversely, no genetic evidence of Polynesians in Native Americans exist, and since the Native DNA was only from a limited group and region, the contact itself is likely quite limited.
APart from this study, the best evidence historically is of course, the sweet potato, which is a crop native to South America, but present in Polynesia before 1492. Even more decisive, as you say, the linguistic connection is present, as it is called a viarant of Kumara both in South America and Polynesia.
So the likely turn of event from this is that a small group of Colombian Native Americans at some point around 1200 made a journey from the Colombian Pacific coast to somewhere in Polynesia, likely the Marquesas Islands. They probably relied on the Southwest going Pacific currents to do that (same currents that the Spanish later used to sail to the Philippines). WIth them they brought sweet potatoes as food (they make excellent ship provisions). Upon arrival, they either found Polynesians, or maybe they settled the islands preceding the Polynesians. Anyway, the encounter must have been peaceful to some extent - the word for sweet potato was adapted, and they interbred, and ahd descendents spreading over Polynesia later.
Some other strands of evidence has been pursued, such as Polynesian chicken bones in Chile, and Polynesian cultural influence on Native American ships, but these have less backing.
The recent study is:
Native American gene flow into Polynesia predating Easter Island settlement, in Nature, it can be read here: