It depends where you look. Some places say Moses is the Hebrew name.
Exodus 2
New King James Version
Moses Is Born
2 And a man of the house of Levi went and took as wife a daughter of Levi. 2 So the woman conceived and bore a son. And when she saw that he was a beautiful child, she hid him three months. 3 But when she could no longer hide him, she took an ark of bulrushes for him, daubed it with asphalt and pitch, put the child in it, and laid it in the reeds by the river’s bank. 4 And his sister stood afar off, to know what would be done to him.
5 Then the daughter of Pharaoh came down to bathe at the river. And her maidens walked along the riverside; and when she saw the ark among the reeds, she sent her maid to get it. 6 And when she opened it, she saw the child, and behold, the baby wept. So she had compassion on him, and said, “This is one of the Hebrews’ children.”
7 Then his sister said to Pharaoh’s daughter, “Shall I go and call a nurse for you from the Hebrew women, that she may nurse the child for you?”
8 And Pharaoh’s daughter said to her, “Go.” So the maiden went and called the child’s mother. 9 Then Pharaoh’s daughter said to her, “Take this child away and nurse him for me, and I will give you your wages.” So the woman took the child and nursed him. 10 And the child grew, and she brought him to Pharaoh’s daughter, and he became her son. So she called his name [a]Moses, saying, “Because I drew him out of the water.”
It could be both. It would be typical of the Bible to infer more than one meaning to a word or name. I've read this other places, but for my own convenience, here is Wikipedia's treatment of this:
"The Egyptian root msy ('child of') or mose has been considered as a possible etymology, arguably an abbreviation of a theophoric name with the god’s name omitted. The suffix mose appears in Egyptian pharaohs’ names like Thutmose ('born of Thoth') and Ramose ('born of Ra'). One of the Egyptian names of Ramesses was Ra-mesesu mari-Amon, meaning 'born of Ra, beloved of Amon'. Ms by itself also has multiple attestations as an Egyptian personal name in the New Kingdom. Linguist Abraham Yahuda, based on the spelling given in the Tanakh, argues that it combines "water" or "seed" and "pond, expanse of water," thus yielding the sense of "child of the Nile" (mw-š).
"The biblical account of Moses' birth provides him with a folk etymology to explain the ostensible meaning of his name. He is said to have received it from the Pharaoh's daughter: "he became her son. She named him Moses [מֹשֶׁה, Mōše], saying, 'I drew him out [מְשִׁיתִֽהוּ, mǝšīṯīhū] of the water'." This explanation links it to the Semitic root משׁה, m-š-h, meaning "to draw out". The eleventh-century Tosafist Isaac b. Asher haLevi noted that the princess names him the active participle 'drawer-out' (מֹשֶׁה, mōše), not the passive participle 'drawn-out' (נִמְשֶׁה, nīmše), in effect prophesying that Moses would draw others out (of Egypt); this has been accepted by some scholars."