Book of Mormon Onomasticon website discusses the etymology of "Josh."
"Josh," Book of Mormon Onomasticon, 2015, accessed February 9, 2023
Etymology
The Book of Mormon PN and GN JOSH may be a hypocoristic form of the HEBREW biblical PN Josiah (Zechariah 6:10, yoʾšiyyȃ, and Jeremiah 27:1, yoʾšiyyȃhû), that is, Josiah minus the theophoric element. Meaning Josiah, and therefore JOSH, could mean “Jehovah has healed,” from the hypothetical HEBREW root *ʾšȃ, “to heal,” posited on the basis of the attested Arabic root ʾsā, “to heal.” On the other hand, Josiah, and thus JOSH, could mean “Jehovah has given,” or “gift of Jehovah,” from the West Semitic root ʾwš, “to give a present" (see Ugaritic ušn, “gift”). As such, JOSH may be related to the biblical period HEBREW PN yʾwš, attested in the Lachish ostraca (where it probably had the pronunciation yāʾuš) and in the Elephantine Aramaic documents. In either case, the Nephite form JOSH, if it does derive from ʾwš or ʾsā, suggests that the aleph must have become quiescent by the time the name appears in the NEPHITE record.
Probably unrelated are the instances of the Amorite names containing ya-(ú-)uš plus a theophoric element, etc., listed in Huffmon, Amorite Personal Names in the Mari Texts (Balimore: Johns Hopkins, 1965), 171. Not only does Huffmon list the root as “uncertain,” but he suggests the preferred etymology be taken from *ġwṯ.
JOSH as a hypocoristicon of Jeshua is somewhat less likely. Ancient Semitic hypocoristica do not function in this manner, i.e., whole lexemes can be dropped, but the lexemes themselves are not subject to being split apart or syncopated. Therefore, JOSH, contrary to the origin of the modern nickname, is not an ancient hypocoristic form of Joshua (from the HEBREW root yehoshua), where Jo- is a shortened version of Jehovah and –shua is from the lexeme for “help; succor.”
Less likely is a derivation from the HEBREW existential particle, yēš, “there is.” It would be hard to account for the /o/ in JOSH from the /ē/ in yēš.
With the following, Kerry Hull discusses a possible Mesoamerican etymology for JOSH. The common Mayan term yax, especially with the meaning “green,” could be a good match for the Book of Mormon GN JOSH. In Mayan languages, the term yax (and its cognates) generally means ‘green’, ‘unripe’, ‘new’, ‘sacred’, and ‘precious’.
The element yax regularly appears as part of ancient Maya GNs recorded in the hieroglyphic script. For example, the famed site of Tikal was known both by the names of Mutal and Yax Mutal (see Tikal Stela 39, AD 376). Many other yax containing GNs are also attested. As a PN, yax is found in PN phrases of the ancient Maya, such as Yax We’en Chan K’inich (Green Eating Sky Great-Sun), a late 8th-century ruler of the site of Xultun. Names of other Maya rulers containing the yax element are very well attested.