What is the basis for the belief that the name of the Messiah is יָהוּשׁוּעַ (Yahushua)?

It stems back to Leah calling her 4th son "Judah". Well, there are no "J's" in Hebrew, which would then bring Judah to Yudah[their letter Yod/Y was converted to J in modern translations]. This Yod/Y is represented as a little apostrophe[the smallest of the Hebrew letters]; you'll notice it above to the right of that Hebrew word for Yahushua. Well Yudah is actually a shortened down form/word that was originally Yahudah. If we break down Yudah or Yahudah, we get[basically] Yah and dah. Yah is what He was called according to Psalm 68:4; and dah means praise...We see this in Gen 29:35. If we take just the Yah or Yahu portion of Judah, and add "shuah", we end up with Yahushua. Why do we add shuah? Shuah is Hebrew for Salvation/Saviour. Jesus could not possibly be enunciated with a "J" sound in Israel 30AD because their alphabet wouldn't allow it. We look at the scriptural renditions of His appellation and attempt to more closely render an English spelling to a Hebraic word uttered years ago; this is called transliteration. Additionally, there is a reliance upon the veracity of the Grecian copies of the original scripts, thus assertion is made that "Yahushua" is the gem.

Links to articles promoting the name Yahushua:

“Originally, the name of the Messiah was pronounced Yahushua. This is the Messiah's original name. When the Gentiles tried to transliterate His name into Greek, they came up with ihsoun or "Iesous". When Iesous was transliterated into Latin, it became "Iesus", which was then carried over into English it became our modern day "Jesus" when the letter "J" developed.”

See article in this link (1): http://www.eliyah.com/nameson.htm

According to the article in the link below (2) Yahushua means Yahuwah Saves or Yahuwah’s Salvation and makes this claim:

“While "Jesus" is commonly used to refer to the Son, it is actually from the Greek "Iesus," which refers to the pagan god, Zeus. The name of Yahuwah’s Son is very much like His own: Yahushua.” The article says that the angel Gabriel said the following to Mary: "You will conceive in your womb, and bring forth a Son, and shall call His name Yahushua." (Luke 1:31) Joseph was told the same thing by the angel: "You shall call His name Yahushua for He shall save His people from their sins." (Matthew 1:21)

(2) https://www.worldslastchance.com/why-yahuwah-a-yahushua-only.html

The notion that the name Jesus means “Hail Zeus” is refuted in this link (3). The article explains how some who say the name of God is YAH, then conclude that the Messiah’s name is YAHSHUA (meaning “YAH Is Salvation”). Part of the article says:

“They contend that is the name used by the apostles and by the Messiah Himself; however, after the apostles were dead and gone, the Roman Church took over Christianity. In order to make their brand of religion more palatable to the pagans, the Roman leaders changed the name of the Messiah into a Greek/Latin hybrid, Iésous, which (supposedly) means “Hail, Zeus.” Since Zeus (or Jupiter) was the chief god in the Greco-Roman pantheon, the pagans had little trouble accepting this new demigod. By changing the Savior’s name, Christianity had been effectively stripped of its Hebrew roots, and the melding with paganism was a success.... the Messiah’s Hebrew name is Yeshua, not Yahshua—the latter being a fabrication in order to make the name sound more like YAH.”

(3) https://www.gotquestions.org/Jesus-Hail-Zeus.html

Finally, an article I found on the Sacred Name Movement (4) illustrates the futility of insisting that one name, and one name alone, is the right name for the Messiah:

“Further, not even the adherents of the Sacred Name Movement can completely agree on what the sacred names actually are. While “Yahweh” and “Yahshua” are the most common, some propose “Yahvah,” “Yahwah,” “Yohwah,” or” Yahowah” for God and “Yeshua” or “Yahoshua” for Jesus. If there is only one non-blasphemous name each for God and Jesus, we better be sure to get it right.”

(4) https://www.gotquestions.org/Sacred-Name-Movement.html

The basis for the belief that the name of the Messiah is יָהוּשׁוּעַ (Yahushua) continues to elude me although Deek's answer is intruiging.