Besides the name Third Reich, the Nazis also used the term Thousand-Year Reich for their period of ruling. Where did they get their inspiration for such a millennium empire?

The references to the 'thousand year Reich' were inspired by, and sometimes also accompanied by refences to the Holy Roman Empire (HRE), which in the eyes of most German historians dated from the coronation of Otto I in 962, and came quite close to lasting a thousand years. It was dissolved in 1806, a few months after Austerlitz. (Some modern English-language works also give 962 as the year of foundation of the HRE, but most opt for the coronation of Charlemagne in 800). The Nazi leadership liked to think they would establish an empire that would 'achieve' what the HRE had failed to do.

All right Bloomsby, seems to be fairly accurate though some sources have the Treaty of Verdun 843
as the beginning of the Holy Roman Empire.
Nonetheless the full-grown YAY goes to Bloomsby, I would think.

German students actually coined 'das dritte reich', it came from the ideological laboratory of the young conservatives. The third reich was the title tract published by Arthur Moeller van den Bruck in 1923, the same year of Hitler's 'Beer Hall Putsch'. The Holy Roman empire of the German nation(1st Reich), Bismark's unified German states(2nd Reich), which ended at the end of WWII followed by Weimar Government.

Third Reich (German: Drittes Reich) denotes the Nazi state as a historical successor to the mediæval Holy Roman Empire (962–1806) and to the modern German Empire (1871–1918). Nazi Germany had two official names, the Deutsches Reich (German Reich), from 1933 to 1943, when it became Großdeutsches Reich (Greater German Reich).
https://en.wikipedia.org/wiki/Third_reich
Which surprised me, I knew about the Holy Roman Empire but thought that the original Roman Empire was considered the 1st Reich as that also lasted around 1000 years (if you count from the founding of the Roman Republic).