In the story of the birth of Moshe (Shemot 2:1-10), several people are mentioned - himself, his parents, his sister, Pharaoh's Daughter, her slaves - but none of these are named. Only at the very end does Moshe receive his new name, and that's the only name given in that story. Why is everyone else nameless in this story?
One possible answer is given by the Tur HaAroch, on why in the posuk וילך איש מבית לוי - “A distinguished man from the house of Levi went, etc.” the name of the "distinguished man from the house of Levi" isn't named at all. Why leave out the name?
The Tur HaAroch explains:
The Torah did not bother to mention Amram’s name at this point, nor that of Yocheved, as the important thing was only to report whose offspring they were. The Torah was anxious at this stage to reveal the birth of the Israelites’ saviour. (emphasis mine)
I have read a shiur in which it is explained why the Torah somewhat cryptically mentions the name of the parents of Moshe Rabbeinu. The shiur is from YuTorah, from the edition "points to ponder, Shemos 5783", written by Rabbi Jonathan Schwartz.
וַיֵּ֥לֶךְ אִ֖ישׁ מִבֵּ֣ית לֵוִ֑י וַיִּקַּ֖ח אֶת־בַּת־לֵוִֽי A man went from the house of Levi (2:1) - Rav Baruch Simon pointed out that we do not identify Moshe’s parents until much later because if we simply noted that he was the son of עמרם we would attribute his existence only to being raised as עמרם’s son and not with his own destiny toward greatness transcending human connection. At the same time, Rav Mendel Blachman added that we DO recognize that Moshe too, had a family in Parshas Voera so that we can know that we too, can have greatness to us as well. Once we know who the great Moshe is, we can also relate to him as a human.
The article also quotes Rav Moshe Feinstein (source unknown):
Only when a parent sees the child reach his or her true potential can the parents take credit for raising him. Even when Moshe stepped up and saved the עברי being beaten by the מצרי, Moshe’s parents’ names are not yet revealed because Moshe did not just merely possess the power of מסירות נפש for one’s fellow man, he also had the power to be Moshe Rabbeinu. Only when Moshe was selected to take Bnei Yisrael out of Mitzrayim do we find the names of Amram and Yocheved mentioned explicitly in the Torah for being able to properly encourage him to achieve his fullest potential
Another chiddush is brought down in the name of Rav Dovid Stav
He notes that except at the beginning of the Parsha, the name Shemos is actually ironic. Fo throghout the Parsha names do not appear prominently. Why? Rav Stav explains that the book of Shemos is really about the building of a nation. A nation does not only require leaders, it requires the Hamon Am – the regular members of the nation prepared to assume their respective roles in creating the nation even if it is without major fanfare. The dedication of the nation to each other is what brings about Geulah.
This goes in line what R' Shmuel Rabinowitz cf. פרשת שמות הני טיים מתוקן (TorahAnytime) says. that the reason why the names are not clearly mentioned, is because people might get the idea that they are not able to achieve greatness, because they are not from "Amram". They are ordinary people, not from a special family, how can they become great and succesful in life? That's, according to R' Shmuel Rabinowitz why the Torah does not mention the names.
Side note: although the OP does not ask why the name Levi is singled out here by the Torah, Rabbi Chaim Tyrer of Tchernovitz, the Be'er Mayim Chaim explains that the reason why the Torah mentions only the name "Levi", e.g. that the "man from the House of Levi" was a descendant of Levi and that would imply that he kept the Torah. He walked in the ways of the Levites, as the Be'er Mayim Chaim says. The Torah saw a need to stress out that Moshe came from this "man from the House of Levi", e.g. the future redeemer of B'nei Yisrael, came from a family that kept the Torah, even in Mitzrayim.
Here's the dvar Torah a Rav (Rabbi David Baadani) just sent me באחד מספריו של הרב פינקוס זצ"ל, which is along the lines of the shiurim brought by Shmuel in his answer:
The Torah teaches us an important lesson here. If it really was written, "And Amram went out and married Yocheved, and she bore a son and named him Moses, and his sister Miriam stationed herself at a distance..." We could think that only a man born to a father who is the greatest of the generation like Amram and a righteous mother like Yocheved, a brother to Miriam who received prophecy at the age of three, Only such a person is capable of being a leader, a savior, a faithful servant. That is why there is no mention of their names in the Torah. So that we know that this is not what made Moshe the leader; but the care, the concern for the individual and the willingness to do everything in his power to alleviate his brother's suffering is the reason he was chosen by G-d to lead the Jewish nation.
Many times what stops us from doing the right thing, reaching out to help, saying a comforting word, giving good advice, is the feeling of inadequacy. 'Who am I? What powers do I have? I'm not Moshe! I was not born to parents like Amram and Yocheved. I just do not fit into it!' G-d tells you, 'It does not matter where you came from. The main thing for me is, where are you willing to go'. The desire, the readiness, the courage to do the right thing, are the ones that give us the power to be an ambassador, a messenger of the Divine. A messenger to spread light, hope and redemption.
לע"נ סבי הרה"ג הרב שמעון בעדני זצוק"ל.
The Ramban addresses the issue of the anonymous man and lady and says the reason the Torah didn't name them was because the Torah would have had to list their lineage, and the shortest possible way to write it was chosen until the birth of the savior. Later their names and lineage were spelled out.
ויקח את בת לוי. ולא הזכיר הכתוב שם האיש ולא שם אשתו אשר לקח, והיה זה בעבור כי יצטרך ליחסם ולהזכיר שמם מי אביהם ואבי אביהם עד אל לוי, ועכשיו ירצה לקצר עד לדת המושיע, ואחר כן (להלן ו יד~כה) ייחס גם שאר השבטים בעבורו בסדר השני: