Hi sencho san,
There is an interesting background from a Chinese legend;
At the Tou (唐) era, a white deer (白鹿) came in the palace of the Emperor Genso (玄宗皇帝; 712-756), and a hermit found it out that the deer had been living for 1,000 years. Examining the deer, a copper plate was found among the white hair around antlers. On the plate, there was an inscription describing the name of a garden to which the deer belonged. The garden existed in the Kan (漢) era and that was 1,000 years before. The Emperor was pleased with this auspiciousness and made a big feast to celebrate the longevity of 1,000 years (千年壽).
This is the origin of the name of the Sake.