... (דברי הימים א ד, יח): וְאֵלֶּה בְּנֵי בִּתְיָה בַת פַּרְעֹה, רַבִּי יְהוֹשֻׁעַ דְּסִכְנִין בְּשֵׁם רַבִּי לֵוִי אָמַר לָהּ הַקָּדוֹשׁ בָּרוּךְ הוּא לְבִתְיָה בַּת פַּרְעֹה, משֶׁה לֹא הָיָה בְּנֵךְ וּקְרָאתוֹ בְּנֵךְ, אַף אַתְּ לֹא אַתְּ בִּתִּי וַאֲנִי קוֹרֵא אוֹתָךְ בִּתִּי, שֶׁנֶּאֱמַר: אֵלֶּה בְּנֵי בִּתְיָה, בַּת יָהּ...
עֲשָׂרָה שֵׁמוֹת נִקְרְאוּ לוֹ לְמשֶׁה: יֶרֶד: חֶבֶר: יְקוּתִיאֵל: אֲבִי גְדוֹר: אֲבִי סוֹכוֹ: אֲבִי זָנוֹחַ. רַבִּי יְהוּדָה בְּרַבִּי אִלְעָאי אָמַר אַף טוֹבִיָּה שְׁמוֹ, הֲדָא הוּא דִכְתִיב (שמות ב, ב): וַתֵּרֶא אֹתוֹ כִּי טוֹב הוּא, כִּי טוֹבִיָּה הוּא.
... (1 Chronicles 4:18) "And these are the children of Bitya, daughter of Pharaoh..." [Said] Rabbi Yehoshua d'Sichnin in the name of Rabbi Levi: The Holy One Blessed Be God said to Bitya daughter of Pharaoh: "Moshe was not your son, and yet you called him your son. So too are you not my daughter, yet I call you my daughter, as it is said: 'These are the children of Bitya...' [rather, read: Bat-YAH]...
Moses was called by 10 names. Yered (the one who goes down); Chever (friend, member, group); Yekuti'el (God will provide/support); Avi G'dor (My father, Fence/Bound in); Avi Socho (My father, his annointed one); Avi Zanoach (My father, he abandons); Rabbi Yehuda in the name of Rabbi Il'Aye said his name was also Tovyah, as it is written (Exodus 2:2) "and when she saw how good (tov) he was," i.e. how good (tovyah) he was;
רַבִּי יִשְׁמָעֵאל בַּר אַמֵּי אָמַר אַף שְׁמַעְיָה שְׁמוֹ. אָתָא רַבִּי יְהוֹשֻׁעַ בַּר נְחֶמְיָה וּפֵרַשׁ הָדֵין קְרָיָא (דברי הימים א כד, ו): וַיִּכְתְּבֵם שְׁמַעְיָה בֶן נְתַנְאֵל הַסּוֹפֵר וגו', שְׁמַעְיָה, שֶׁשָּׁמַע יָהּ תְּפִלָּתוֹ. בֶּן נְתַנְאֵל וגו' בֵּן שֶׁנִּתְּנָה לוֹ תּוֹרָה מִיַּד לְיַד. הַסּוֹפֵר, שֶׁהָיָה סוֹפְרָן שֶׁל יִשְׂרָאֵל. הַלֵּוִי, שֶׁהָיָה מִשִּׁבְטוֹ שֶׁל לֵוִי. לִפְנֵי הַמֶּלֶךְ וְהַשָּׂרִים, לִפְנֵי מֶלֶךְ מַלְכֵי הַמְּלָכִים הַקָּדוֹשׁ בָּרוּךְ הוּא וּבֵית דִּינוֹ. וְצָדוֹק, זֶה אַהֲרֹן הַכֹּהֵן. וַאֲחִימֶלֶךְ, שֶׁהָיָה אָחִיו שֶׁל מֶלֶךְ. בֶּן אֶבְיָתָר, שֶׁוִּתֵּר הַקָּדוֹשׁ בָּרוּךְ הוּא עַל יָדָיו מַעֲשֵׂה הָעֵגֶל.
(Leviticus Rabbah cont.) Rabbi Yishmael Bar Amei said his name was also Shemayah--Rabbi Yehoshua Bar Nechemyah comes and explains why [it can be claimed that he] was called thus. Scripture teaches: (1 Chronicles 24:6) "Shemayah son of Netanel, the scribe, the Levite, registered them..."--"Shemayah," for YAH heard his prayer, "Son of Netanel," for he was the son to whom God gave the Torah from hand to hand, "the Scribe," for he was the scribe of Israel, "The Levite," for he was of the tribe of Levi, "before the King and his ministers," before the King of Kings of Kings, the Holy One Blessed Be God and his Heavenly Court, "And Zadok," this is Aharon HaCohen, "And Achimelech," for he (Moshe) was the brother of a king, "Son of Evyatar," for the Holy One Blessed Be God relented (viteir) on Moshe's behalf during the incident of the Golden Calf.
רַבִּי תַּנְחוּמָא בְּשֵׁם רַבִּי יְהוֹשֻׁעַ בֶּן קָרְחָה אָמַר אַף לֵוִי הָיָה שְׁמוֹ עַל עִקַּר מִשְׁפַּחְתּוֹ (שמות ד, יד): הֲלֹא אַהֲרֹן אָחִיךָ הַלֵּוִי. וּמשֶׁה, הֲרֵי עֲשָׂרָה, אָמַר לוֹ הַקָּדוֹשׁ בָּרוּךְ הוּא לְמשֶׁה חַיֶּיךָ מִכָּל שֵׁמוֹת שֶׁנִּקְרְאוּ לְךָ אֵינִי קוֹרֵא אוֹתְךָ אֶלָּא בַּשֵּׁם שֶׁקְּרָאַתְךָ בִּתְיָה בַת פַּרְעֹה (שמות ב, י): וַתִּקְרָא שְׁמוֹ משֶׁה, וַיִּקְרָא אֶל משֶׁה.
Rabbi Tanchuma in the name of Rabbi Yehoshua ben Karcha said that also Levi was his name, essentially as a reference to his family: (Exodus 4:14) "There is your brother Aaron the Levite..." And Moshe, which makes ten. The Holy One Blessed Be God said to Moses: Though all the names you are called may bring you joy, I will only call you by the name you were called by Bitya daughter of Pharaoh: (Exodus 2:10) "And she called his name Moshe...", And God called to Moses.