Polynesian navigation methods were pretty sophisticated, especially considering the low level of technology available to them. Not only did they have excellent knowledge of star maps in both hemispheres (impressive enough for a non-literate culture that had to pass this information by word of mouth) they also paid attention to things that European navigators usually ignored, such as cloud patterns, the character of waves, the colour of the water and the 'wrack' (e.g. the small pieces of debris carried on waves) to determine how close to land they were and which land area it was.

They could definitely survive at sea for long periods of time. In addition to techniques for collecting rainwater, their canoes (known in Maori and Hawaiian as 'waka') had lots of room for food. Some cultures used double hulled canoes to increase survivability and storage space while on long sea voyages.

Their system wasn't perfect. Although it's believed that most of the Polynesian settlements were deliberate, sometimes one Polynesian island group did become isolated because long distance navigation methods were lost. Probably the best example is the Maori of NZ, who arrived as colonists but forgot the methods for returning to their point of origin. Of course, given that New Zealand was a much bigger and more fertile landmass than where they'd come from, they didn't really need to go back, so the knowledge was probably not retained because it wasn't useful. I believe this is the case for the Hawaiians, too.

There is a pretty strong indication that Polynesians did actually reach South America - the most persuasive is the widespread Polynesian cultivation of the sweet potato (called 'kumara' or something similar in most Polynesian languages) which originates from South America. But it wasn't a good place for settlement, so contact seems to have been limited to some trade, which was discontinued when the trade goods (for example, kumara) could be obtained domestically.

So in summary, given their lack of navigation technology (compasses, telescopes), Polynesian navigation was very advanced, and it's not surprising that they managed to colonise most of the Pacific.

Source: Michael King, 'New Zealand: Its land and its people'.

They probably didn't.

Sweet potatoes, a south american crop, are found in Polynesia, and chicken, an Asian bird, is found in south america, both pre-contact.

Also, Polynesia DNA is found in south america.

I think you mean Polynesia, not Melanesia.

I don't know that they missed it, it's just that by the time they might have gotten there, it was already populated. The Pacific Islands were effectively the last places humanity colonized.^1 You mention Easter Island- the estimates for its colonization by humans at the earliest is 300 AD, maybe even as late as 1200. Hawaii itself was populated no earlier than 800 AD. But the Americas were populated by 15,000 years ago, possibly as early as 30,000 years ago. So it's entirely possible that Pacific Islanders did reach the Americas, only to find people that people had already been there for well over 10,000 years.

If they did land in the Americas, they might have been brought in to the culture, in which case there might be genetic remnants. Or they might have been killed as invaders. Or they might have turned back on finding it already populated. Or they might have not ever made it all.

But we can say for certain that the reason they weren't the ones to populate the Americas is because by the time they might have gotten there, the indigenous people of the Americas were already very well established.

1.: And it's not like they found all the islands in the pacific, anyway. This is a map of the places still uninhabited by the European Age of Exploration: http://www.slate.com/articles/news_and_politics/map_of_the_week/2013/08/map_of_discoveries_of_european_explorers.html