The name Sh’mini Atzeret, “8th Day of Solemn Assembly”, is from Num. 29:35, Bayom hash’mini, atzeret tih’yeh lachem – “On the 8th day you shall have an atzeret“. The day is also called atzeret in Lev. 23:36.
The translation “solemn assembly” is rather peculiar. The Hebrew might possibly be rendered “assembly”, but why add the adjective “solemn”?
The root of atzeret is a-tz-r, which usually means to cease or desist, with the extended meaning of to arrest. The 8th day is fundamentally a concluding festival that celebrates the end of the Tishri cycle of holydays.
It has no intrinsic observance – nothing like matzot on Pesach and the lulav on Sukkot: prayers for rain were a later development.
What then is its purpose?
Another verse provides an answer when it says that we should celebrate for seven days and be ach same’ach, “altogether joyous” (Deut. 16:15), so Sh’mini Atzeret is as it were a celebration of the blessing of being able to celebrate Sukkot.
However, it is not counted as a day of Sukkot but is regel bif’nei atzmo – “a festival on its own” – with Simchat Torah as its 2nd day (except in Israel, where Sh’mini Atzeret and Simchat Torah are one and the same day).
The prayers for rain should technically have been said on the first day of Sukkot, but they were postponed until Sh’mini Atzeret because the halachah was sensitive to the needs and feelings of the Jewish people.
On the first day people might not have prayed so sincerely for rain because it might spoil the meals in the sukkah.