Lewis and Clark trekked up the Mississippi river in 1804 and spent the winter of 1804-05 at Ft. Mandan (present day Washburn, North Dakota). Lewis believed that the Mandan people were descended from Madoc ap Owain Gwynedd, who purportedly sailed from the new world in 1170 after the death of his father, and to escape the murder and infighting among his brothers for the throne of Wales. Given that all but one of his brothers ended up dead within 5 years, this might have been a good plan, all around.
Now, if Madoc’s family hadn’t been associated with the Danes of Dublin, the notion of such an expedition would have been even more far-fetched. Madoc’s great-grandmother was Ragnhild, “the daughter of Olaf of Dublin, son of King Sigtrygg Silkbeard and a member of the Hiberno-Norse Uí Ímhair dynasty. Through his mother, who appears in the list of the fair women of Ireland in the Book of Leinster.” His grandfather, Gruffydd, “claimed relationships with many of the leading septs in Ireland. His great-great grandparents on his mother’s side include the High King of Ireland, Brian Bóruma, and the King of Dublin and King of Northumbria, Olaf Cuarán, and Gormflaith.” (Hudson, Benjamin T. (2005). Viking Pirates and Christian Princes: Dynasty, Religion, and Empire in the North Atlantic (Illustrated ed.). United States: Oxford University Press). The norsemen had landed in North America over a hundred years before, stories about which Madoc would have heard.
Welshmen were not the same kind of sailors as the Danes, but one of the pieces of evidence that Meriwether Lewis and others (George Carlin) hit upon that made him think that the Mandans were indeed descended from Welshmen (in addition to their anomalous hair and eye color), were their coracles, round boats not unique to Wales, but used in the British isles for thousands of years and not found elsewhere in North America. See http://www.data-wales.co.uk/coracle1.htm for a description. And here for a long discussion about this issue: http://freepages.family.rootsweb.ancestry.com/~bowen/mandans.html
That this story is anything more than a myth remains unlikely. The origin of the story is not known, but it rose to prominence during the Elizabethan era, when England was competing with Spain for the conquest of the New World. That a Welsh explorer had come to America first . . . well, that would strengthen the claim. That doesn’t mean it didn’t happen, but as with Arthur, propaganda is a powerful tool. “The earliest certain reference appears in a cywydd by the Welsh poet Maredudd ap Rhys (fl. 1450-83) of Powys, which mentions a Madog who is a son or descendant of Owain Gwynedd and who voyaged to the sea. The poem is addressed to a local squire, thanking him for a fishing net on a patron’s behalf. Madog is referred to as “Splendid Madog… / Of Owain Gwynedd’s line, / He desired not land… / Or worldy wealth but the sea.” http://en.wikipedia.org/wiki/Madoc
The Mandans as a people, did not survive the 19th century and there is no other evidence that might help us. “The great plague of smallpox struck the Three Tribes in June of 1837, and this horrible epidemic brought disaster to these Indians. Francis A. Chardon’s journals state that on July 14, a young Mandan died of smallpox and several more had caught it. The plague spread with terrible rapidity and raged with a violence unknown before. Death followed in a few hours after the victim was seized with pain in the head; a very few who caught the disease survived. The Hidatsa scattered out along the Little Missouri to escape the disease and the Arikara hovered around Fort Clark. But the Mandan remained in their villages and were afflicted worst; they were afraid of being attacked by Sioux if they ventured out of their villages. By September 30, Chardon estimated that seven- eighths of the Mandan and one-half of the Arikara and Hidatsa were dead. Many committed suicide because they felt they had no chance to survive. Nobody thought of burying the dead, death was too fast and everyone still living was in despair. The scene of desolation was appalling beyond the conception of the imagination. The Mandan were reduced from 1800 in June to 23 men, 40 women, and 60 to 70 young people by fall. Their Chief Four Bears, had died. (Shane, 1959, p. 199).” http://freepages.family.rootsweb.ancestry.com/~bowen/mandans.html