Moses had quite a few names, but this name is from his Egyptian name. "When the child matured, [his mother] brought him to Pharaoh's daughter. She adopted him as her own son, and named him Moses (Moshe / משה). 'I bore (mashe) him from the water,' she said.” (Sh'mot / Exodus 2:10).
There were even a number of Pharaohs with similar names including Thutmose who was the third pharaoh of the 18th dynasty of Egypt. He received the throne after the death of the previous king, Amenhotep I.
Some of our sages (including חזקוני / Chizkuni of the13th century) state that his mother named him and told the Egyptian princess the name...
The footnote from The Living Torah says: "In Egyptian, Moshe means a son. Thus, his naming is prefaced by a phrase that is literally translated, 'he became to her as a son' (cf. Ibn Ezra; Hadar Zekenim). Significantly, the suffix moshe is found (and exclusively so) in the names of many Pharaohs of the 18th Dynasty, such as Ka-moshe ('son of [Ra's] majesty'), Ach-moshe (Ahmose; 'son of the moon,' or 'the moon is born') and Toth-moshe (Thutmose; 'son of Toth'). The word moshe may indeed be of Semitic origin (see next note, this verse, 'bore'), introduced by the Semitic Hyksos.
"According to other ancient sources, the name Moses comes from the Egyptian mo (water) and uses (drawn from) (Josephus, Antiquities 2:9:6, Contra Apion 1:31; Philo De Vita Moses 2:17; Malbim)."
Some sources state that Moses' Egyptian name was Monius (Ibn Ezra; cf. Abarbanel; Josephus, Contra Apion 1:26, 28). Other ancient sources claim that Moses' name was preserved among the Gentiles as the legendary Musaeus, teacher of Orpheus, from whom the Muses obtained their name (Artapanus, in Eusebius, Preparatio Evangelica 9:27).
Artscroll's footnote says: "She gave him the Egyptian name Monios, which means that he was drawn from the water. Moses/Moshe is the Hebrew translation of that word (Ibn Ezra)."
Vayikra Rabbah 1:3 (midrash probably compiled around the fifth century CE) tells us that Moses had ten names:
Miriam (his sister) gave him the name Yered (ירד).
His grandfather called him Avigdor (אבי גדור).
Some say his father named him Chever (חבר).
His mother called him Yekutiel (יקותיא‑ל).
His brother Aaron called him Avi Zanoach (אבי זנוח).
Another name was Tuviah (טובי‑ה).
The Jewish people named him Shemayah (שמעי‑ה).
Ben Evyatar (בן אביתר) and finally
From Vayikra Rabbah 1:3:
The Holy One said to Batya, the daughter of Pharaoh: "Moses was not your son, and yet you called him your son. So too, you are not my daughter, but I call you my daughter."
Moses was known by 10 names: Jered, Chever, Yekutiel, Avigdor, Avi-Socho, Avi-Zanoch ... Tuvia ... Shmaya ... Levi ... and Moses, which makes 10.
The Holy One said to Moses, Behold! From among all the names from which you are known, I only refer to you by the name that Batya, Pharaoh's daughter, named you.
1. Yered (ירד), implying descent. According to one opinion, Miriam gave him this name, for because of him, she went down (yarad) to the Nile to see what would become of him. Alternatively, Moses was called this name because he brought the Torah down to the Jewish people, and the Divine Presence back down to this physical world.
2. Avigdor (אבי גדור), master of the fence. According to the Yalkut Me'am Loez, he was called this (by his grandfather, Kehat), because "since Moses' birth, G‑d has fenced in Pharaoh, not allowing him to continue his decree to drown Jewish infants."
3. Chever, (חבר) meaning, companion, or connector. Either because Moses connected the Jewish people with their heavenly Father, or because he prevented Heavenly retribution for their sins. Some say that Amram, his father, gave Moses this name, because Moses was born after his father had once again joined his wife after having divorced her.
4. Avi Socho, (אבי סוכו) Father of Seers. He was given this name by his grandfather, Kehat (alternatively, by the nurse who helped Moses' mother raise him), because Moses would grow up to be the "master" (avi) of the seers (sochim) and prophets.
5. Yekutiel (יקותיא-ל), from the root kavei (קוה) meaning hope. His mother Jochebed called him this name because she had hope and trust in G‑d that He would return Moses to her. Alternatively, because she foresaw that Moses would be the Jewish nation's great hope.
6. Avi Zanuach (אבי זנוח), literally, "master of rejection." Aaron, Moses' brother gave him this name, saying "My father rejected my mother, but took her back because of this child." Alternatively, because Moses would make Israel reject idols.
7. Tuvia (טובי'ה), implying goodness.
8. The Jewish people called him "Shemaya (שמעי-ה) ben [the son of] Nethanel." They predicted that in his days, G‑d would hear (שמע) their prayers.
9. Ben Avitar (בן אביתר), son of pardon, since Moses was the Jewish son who would solicit G‑d's pardon (ויתר) for the Jewish people's sin of the Golden Calf.
10. Levi (לוי), so named after the tribe to which Moses belonged.
(Source: Rabbi Eliezer Danzinger, Chabad.org)