I can think of three that might vie for the honour.
Welsh - where the days of the week are still modelled on the original Indo-European pattern.
Sami (spoken by the people formerly known as Lapplanders) - this is Finno-Ugric, so similar to the languages spoken in Finland and Hungary and ultimately coming from the Asian steppes, but it is a distinct language and has been preserved relatively unchanged by isolation.
Top of my list, however, would be Basque - it has no 'sister'languages anywhere in the world, as far as I know, so in a sense it has no historical "start point".