I've read that there is some linguistic relationship between the Basque language and at least one of the American Indian languages. Have you ever heard anything along that line?
Merritt Ruhlen thinks so. He calls it the Dene-Caucasian family. Basque is believed to be distantly related to languages such as Chechen in the Caucasus. Ruhlen thinks those are related to some Siberian languages and to the Athapaskan Indian languages of the Far North and the Southwest (Navajo, Apache, and the Dene languages of Alaska.)
Some people have thought that Basque is directly descended from the languages spoken in Upper Paleolithic Europe, because the Basques are more directly descended from Cro-Magnon man than other Europeans, although the Celtic-speaking peoples are genetically close to the Basques (The origins of Celtic languages are pretty obscure, but it would seem that they originated in central or eastern Europe rather than Britain, Ireland, or Western France.)
Others tend to see languages as being related to gene markers carried by the original speakers (e.g. Spencer Wells) and dismiss the notion of related languages being associated with genetically very different peoples. Have you ever read Journey of Man? It's based on a PBS special that aired last fall, I think.
Wells thinks that Basque is part of a set of languages carried by the earliest farmers to migrate out of the Middle East. These, according to him, may include the languages of the Caucasus, Burushaski (a language spoken by Shiite Muslims in Pakistan), Sumerian, the Etruscan and Pelasgian languages spoken in the Mediterranean before the Greeks and Romans, and Iberian languages of Spain.
He correlates this distribution of languages with the frequency of a genetic marker called M172, which occurs most often in the Caucasus, the Middle East, and the eastern Mediterranean (Greece,Italy,and southern Balkans).
But the Basques have very low frequencies of M172; their marker is M173 (the Upper Paleolithic European marker). So there could be no connection, and Basque really is a relic of the languages spoken in Europe during the last Ice Age, and truly is a linguistic isolate.
OTOH, the Celtic languages are spoken by people who are genetically closer to the Basques than to the original Celtic speakers, so maybe the Basques adopted Basque from the first farmers to migrate into Spain, just like the Irish and Scots adopted Celtic from elites originating in continental Europe (probably in the early Bronze Age- remember the Amesbury archer?) who left little or no genetic trace in the extreme west of Europe.
Wells favors the Kurgan hypothesis (Gimbutas) for Indo-European origins, so he doesn't think IE languages correlated with the first farmers in Europe (why he thinks M172 is associated with a mostly extinct substratum of languages). Recent evidence (I'll post it when I find it) from using new methods to analyze language evolution supports the Wave of Advance model (Renfrew, Gamkrelidze-Ivanov that says first farmers = proto Indo-European speakers.
It's an open question; you can't talk about the origin of the Basques without touching on the origin of IE languages.