The name "Jesus" in English has a complicated linguistic history that isn't apparent in modern Bibles.
"Jesus" is an Anglicized form of the Greek name Yesous found in the New Testament. Yesous represents the Hebrew Bible name Yeshua, which occurs as "Jeshua" in English Bibles (Ezra 2:2; Neh 7:7). In Medieval English the "J" was pronounced as a "Y."
Yeshua, in turn, is a shortened form of the name Yehoshua ("Joshua" in English Bibles).
Moses' right-hand man, Joshua, has three names in the Bible. Originally, it was Hoshea, but Moses changed it to Yehoshua (Num 13:16). During the Babylonian Exile, it was shorted to Yeshua (Neh 8:17).
"Yehoshua" is a compound name consisting of two elements.
(1) The prefix "Yeho–" is an abbreviation of the Tetragrammaton, God's Four-Letter Name: Yod-He-Vav-He: YHVH. (Modern scholars think the third letter was pronounced as "W." Thus: YHWH, Yahweh.)
The "Name" YHVH is used over 6,800 times in the Hebrew Bible. It appears in most English Bibles printed with large and small capital letters: LORD. This stylized euphemism invented by Medieval printers distinguishes it from the words Adon and Adonai, both translated "Lord."
In the Hebrew Bible "Yeho-" is a prefix form of God's name. It's used at the beginning of certain proper names: Jehoshaphat, Jehoiachin, Jehonathan (the "J" was pronounced as "Y" in Medieval English).
The Tetragrammaton also has a suffix form that occurs in some names: "-yah." In the Greek Old Testament (Septuagint) this ending is spelled "-iah" and appears in English Bible names such as Isaiah, Jeremiah, Nehemiah, and Zechariah. Elijah is Eli-Yah, my God is YHVH.
(2) The second element of the name Yeho–shua is a form of the Hebrew verb yasha which means to deliver, save, or rescue.
Thus, linguistically, the name "Yehoshua—Yeshua—Jesus" conveys the idea that God (YHVH) delivers, saves, rescues — eventually through his servant Messiah, who bears God's name.
The Hebrew Bible word "savior" [moshiah] is also rooted to yasha.
Moshiah is used 9x for God (2 Sam 22:3; Isa 43:3; 45:15, 21; 49:26; 60:16; 63:8; Jer 14:8; Hos 13:4). It's used 5x for human "deliverers," "rescuers" or "saviors" (Judg 3:9, 15; 2 Kgs 13:5; Obad 1:21; Neh 9:27).
Isaiah 19:20 may refer to the Savior-Messiah: "He [God] will send [to the Egyptians] a Moshiah and a Rav [champion] and he will deliver [natzal] them."
God Gives His Name to People
After the "Aaronic Blessing" is pronounced over the people of Israel (Numbers 6:24-26), God tells Aaron and his sons: "So they shall put [sim] my Name on the children of Israel" (v. 27). This act becomes a key identity marker in the future.
In the book of Jeremiah, God gives his name both to the future son of David (the Messiah) and to Jerusalem. King and people are owned by, belong to, share the divine Name of their God.
I shall raise up for David a righteous Branch [or Son]...
Accordingly, in Matthew 1:21 "Yeshua" is the birth-name God gave his Son: "YHVH saves."
She will bear a son, and you shall call his name Yesous
Also note the play on words in the name "Yeshua" and the noun "yeshuah" that Hebrew speakers would hear in Acts 4:
There is salvation [Heb. yeshuah] in no one else;
For more details on the Hebrew behind the name "Jesus Christ," see Shem Yeshua Mashiach.
HaYachid—The Unique Messiah [PDF]