I wonder if DNA testing might shed some light? Could one test samples of indigenous native American peoples and compare them to those of the European areas whence the supposed migrants originated?
I remember some interesting scientific data about the inhabitants of Orkney and Shetland and the Vikings. It appeared to show that most of the present Scots up there have no Viking DNA at all. My late father would have been disappointed. He came from the far north, and always swore he had Scandinavian blood (very blond and blue-eyed) from his 'Viking ancestors'.
Given the navigational and sea-faring skills of the Polynesians for a start, it's not impossible to imagine they crossed vast stretches of ocean.
I think the artefacts found (as mentioned in the link) are most interesting, with their differences to known types of tool from Native American finds.