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" is now.
Yeshua, in turn, is a shortened form of the name Yehoshua ("Joshua" in English Bibles).
Moses' righthand 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.)
In the Hebrew Bible "Yeho-" is used at the beginning of certain proper names: Jehoshaphat, Jehoiachin, Jehonathan (the "J" was pronounced as "Y" in Medieval English). The suffix form of the Tetragrammaton is "-yah" ("-iah" in Greek, as in Isaiah, Jeremiah, Zechariah, or Halleluiah).
(2) The second element of the name Yehoshua 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 or saves (his people), eventually through his servant messiah.
What is not evident outside the Hebrew strata is that the word "savior" 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) and 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 rigteous Branch [or Son]...
Accordingly, in Matthew 1:21 "Yeshua" is the name God gave his Son. It is his original birth-name.
"She will bear a son, and you shall call his name Yesous [Greek; Yeshua in Hebrew], for it is he who will save his people from their sins."
Now note the play on words on the name "Yeshua Messiah" that Hebrew speakers would hear in Acts 4:
"There is salvation [Heb. yeshuah] in no one else; for there is no other Name under heaven that has been given among men by which we must be saved [Heb. yasha]" (v. 12).
For more details on the Hebrew behind the name "Jesus Christ," see Shem Yeshua Mashiach.
For a good article on the name "Jesus" see One For Israel's "Jesus vs. Yeshua?"