The name Shuah in the Bible
There are four different Hebrew names that transliterated into English form the name Shuah, or variations thereof depending on the translation. The name שוע, Shoa (or variations thereof) is spelled the same as one of the Shuahs but pronounced slightly different.
The first Shuah (שוח; pronounced shuach) is a son of Abraham with Keturah (Genesis 25:2). A famous descendant of this Shuah (we assume) is Bildad the Shuhite (שוחי), one of the four friends of Job (Job 2:11). A feminine variation of this name occurs in 1 Chronicles 4:11: שוחה (Shuhah; pronounced shuachah), which is assigned to a (female?) descendant of Judah.
A completely different name occurs in Genesis 38:2,12: שוע (pronounced shua'), where it is the name of the father of a wife of Judah (perhaps also known as Bathshua — Genesis 38:12). A variant of this name occurs in 1 Chronicles 7:32: שועא (pronounced shua'a), where it is assigned to a daughter of Heber, a granddaughter of Asher, son of Jacob.
The name Shoa (שוע) occurs in Ezekiel 23:23, where it is the name of a Chaldean tribe.
Etymology of the first two names Shuah
The names שוח and שוחה come from the verb שוח (shuah), denoting a motion towards a low position:
The name of this son of Abraham (Genesis 25:2) and the name of this perhaps female descendant of Judah (1 Chronicles 4:11) mean Brought Low or Put In A Pit. Jones' Dictionary of Old Testament Proper Names reads Prostration.
Etymology of the second two names Shuah and Shoa
The names שוע and שועא look like they have something to do with the verbs שוע (shawa), to cry out (for help), and ישע (yasha'), meaning to be saved or delivered:
Shuah and Shoa meaning
BDB Theological Dictionary lists the Shoa as mentioned in Ezekiel 23:23 under the root שוע (shawa), meaning to cry. This name thus means A Cry.
BDB Theological Dictionary refers both versions of this form of Shuah to the verb ישע (yasha'), meaning to be saved. The names of the father of Judah's wife (Genesis 38:2) and the Asherite woman (1 Chronicles 7:32) may therefore mean Noble.
However, there is a peculiar noun that is used in Job 30:24 and 36:19, that is spelled שוע. Some translations take this word to mean 'riches' (NAS, KJV), although BDB Theological Dictionary says that a derivation from שוע (shawa), to cry, is preferred, and HAW Theological Wordbook of the Old Testament doesn't even list the alternative of riches.
Still, NOBSE Study Bible Name List reads Rich for Shoa and Prosperity for all variations of Shuah. Jones' Dictionary of Old Testament Proper Names reads Wealth for Shua.