Pesach was Passover, Shavu’ot was Pentecost, Rosh HaShanah was New Year, Yom Kippur was The Day of Atonement, Sukkot was Tabernacles… and Sh’mini Atzeret was The Eighth Day of Solemn Assembly.
This latter name sounded highly impressive, but no-one really knew what it meant.
They could have looked up the Chumash and they would have found the combination of the words sh’mini and atzeret (Num. 29:35; cf. Lev. 23:36) but the two words are not a complete phrase (“sh’mini atzeret”) but part of a sentence, “On the 8th day you shall have an atzeret”.
The root of the word is ayin-tzadde-resh, which means to stop or close.
It is used in relation to Shavu’ot which is the atzeret, the closing festival of Pesach, occurring after seven weeks of the Omer. By analogy, Sh’mini Atzeret is the closing festival of Sukkot.
In theory it should have been seven weeks after the beginning of Sukkot, but that would have been well into the winter season when a pilgrimage to Jerusalem would have been difficult.
To make life easier for the Israelites, says the Midrash, God moved the closing festival and made it right at the end of the Sukkot celebrations.