While there is nothing explicit given regarding the change, the significance appears to lie in the meanings themselves. However, this topic is possibly the most important onomastic study of all time. No exaggeration.
Numbers 13:16 reads: “אֵלֶּה שְׁמֹות הָאֲנָשִׁים אֲשֶׁר־שָׁלַח מֹשֶׁה לָתוְּר אֶת־הָאָרֶץ וַיִּקְרָא מֹשֶׁה לְהֹושֵׁעַ בִּנ־נוְּן יְהֹושֻׁעַ”
First, we must identify the original name, phonologically speaking. The vocalizations (vowel markings) reflected above date to about 850 C.E., when the Masoretic system of vowel pointing was completed. These merely preserved the pronunciations from the reading tradition then practiced in the synagogues. Sometimes, we can do better. The original form of the name /Hôšēaʿ/ (Hosheaʿ) was /Hawšēʿ/ before contraction of /aw/ > /ô/. The ʿayin /ʿ/ is a pharyngeal (throaty) sound. The patach furtive ( ַ ) is there to assist with these sounds. This original spelling is confirmed from the eight century B.C.E., where Hoshea, King of Israel, is written in cuneiform as a-ú-se-aʾ in 732 B.C.E. (S. Cole, “Awsēa’” in The Prosopography of the Neo-Assyrian Empire [ed. K. Radner; Helsinki: University of Helsinki, 1998], 1:238). The [aw] is pronounced as in “auto” not as in “cow.” This is the so-called Hebrew diphthong. The Septuagint (c. 250 B.C.E.) also confirms the ancient pronunciation, where Num 13:16 reads: “ταῦτα τὰ ὀνόματα τῶν ἀνδρῶν, οὓς ἀπέστειλεν Μωυσῆς κατασκέψασθαι τὴν γῆν. καὶ ἐπωνόμασεν Μωυσῆς τὸν Αυση υἱὸν Ναυη Ἰησοῦν.” Again, the spelling Αυση reflects a Hebrew *הַוְשֵׁע.
The form *הַוְשֵׁע indicates a causative imperative masculine singular of the verb wšʿ (> yšʿ) [IPA wʃʕ]. It means “Save!” or “Oh Save!” This was the meaning of the original name. But in context this would apply to the events before the Exodus, with the oppression and a cry: “וַיְהִי בַיָּמִים הָרַבִּים הָהֵם וַיָּמָת מֶלֶךְ מִצְרַיִם וַיֵּאָנְחוְּ בְנֵי־יִשְׂרָאֵל מִנ־הָעֲבֹדָה וַיִּזְעָקוְּ וַתַּעַל שַׁוְעָתָם אֶל־הָאֱלֹהִים מִנ־הָעֲבֹדָה” (And in those many days the King of Egypt died and the children of Israel sighed because of the labor and they cried, and their cry went up to Elohim because of the labor, Exo 2:23). The response was given in 3:8, “וָאֵרֵד לְהַצִּילֹו מִיַּד מִצְרַיִם וְּלְהַעֲלֹתֹו מִנ־הָאָרֶץ הַהִוא אֶל־אֶרֶץ טֹובָה וְּרְחָבָה” (I have come down to deliver them from the hand of the Egyptians and to bring them up from that land to a good and broad land). That name seemed to typify the sentiments in bondage.
Moses is written as performing the change while at Kadesh, in the wilderness of Paran, when they went to spy out this land of Canaan. He introduced the Israelites’ Supreme Being into it, Yahweh. He changed *הַוְשֵׁע (Hawšēʿ) to יַהְוְשֻׁע (Yahwšūʿ), but not יְהֹושֻׁעַ (Yĕhôšūʿ) or יֵשׁוּעַ (Yēšûʿ). These latter two trace to the later Hellenistic period and exemplify religio-linguistic machinations designed to avoid a form too close to Yahweh, i.e. yahw—the only long pre-exilic abbreviated form of the name at the end and also at the beginning of personal names compounded with the name Yahweh. For the abbreviated form yahw, see David Noel Freedman and Michael P. O’Connor, “יהוה YHWH,” in Theological Dictionary of the Old Testament, ed. G. J. Botterweck and H. Ringgren (Grand Rapids: Eerdmans, 1986), 5:501, 512 and David Noel Freedman, Francis I. Anderson, and A. Dean Forbes, Studies in Hebrew and Aramaic Orthography (Winona Lake: Eisenbrauns, 1992), 173–74. Also from the Akkadian side see Ran Zadok, “Old Iranian Anthroponyms and Related Material in Late Babylonian Sources,” Revue d’assyriologie et d’archéologie orientale 98 (2004): 7 n. 7; and “Jehu,” Nouvelles Assyriologiques Brèves et Utilitaires 1 (1997): 20. Regarding Yĕhô, William Albright commented (rather strongly) that: “In the same way they [Masoretes] found the spelling יהו with the pronunciation Yô. There was one way out of the difficulty—to point the initial yôd with šewâ, which they promptly did. . . . At all events, their system forced them to create an anomalous punctuation which presently became a literary pronunciation, and has been responsible for gallons of wasted ink in recent times.” (William F. Albright, “Further Observations on the Name Yahweh and Its Modifications in Proper Names,” Journal of Biblical Literature 44 (1925): 160). Albright refers to the Greek transcriptions of the initial element exclusively with Ιω, with no [e] before the ω. Nevertheless, while the previous authors understood the original correct form (yahw), it is not certain how they understood the precise chronological development. The spelling of compound names with Ιω is a regular late ideological change after the fourth century B.C.E., but יֵשׁוּעַ (Yēšûʿ) (with no abbreviated name) is an anomaly witnessed already at Mt. Gerizim at about 200 B.C.E. See Y. Magen, H. Misgav, and L. Tsfania, Mount Gerizim Inscriptions: Volume I The Aramaic, Hebrew and Samaritan Inscriptions (Jerusalem: Israel Antiquities Authority, 2004), 20. Our own recent investigation has greatly clarified the development, but must be left for another question. See my preliminary studies here and ideological developments about Yahweh here. For now, we can say positively that the form Yahwšūʿ is the original form of this name and meant “Yahweh is salvation,” but has a very long history.
Thus he changed Hawšēʿ (Oh Save!) to Yahwšūʿ (Yahweh is salvation). It seems that Moses utilized the change to remind the Israelites just before they were to enter the land of their moment of distress at the Red Sea when Moses said “אַל־תִּירָאוְּ הִתְיַצְבוְּ וְּרְאוְּ אֶת־יְשׁוְּעַת יְהוָה אֲשֶׁר־יַעֲשֶׂה לָכֶם הַיֹּום” (Fear not, stand still, and see the salvation of Yahweh that he will work for you today). Yahweh did save them, as seen in Exo 14:30 “וַיֹּושַׁע יהוה בַּיֹּום הַהוְּא אֶת־יִשְׂרָאֵל מִיַּד מִצְרָיִם וַיַּרְא יִשְׂרָאֵל אֶת־מִצְרַיִם מֵת עַל־שְׂפַת הַיָּם” (And Yahweh saved Israel in that day from the hand of the Egyptians and Israel saw the Egyptians dead upon the edge of the sea).