Seeing as the previous Pasuk stated "Nasi Echad la'Yom, Nasi Echad la'Yom", why does the Torah insert the word "ba'Yom ha'Rishon"?
Sifri: To teach us that Nachson ben Aminadav brought his Korban on the first day of the week. 1
See Torah Temimah, note 9. Refer also to 7:48:151:1.
Why does the Torah not mention "Nasi" with regard to Nachshon ?
Da'as Zekeinim: To prevent him from becoming conceited, for being the first to bring his Korban. 1
Gur Aryeh: To teach us that he did not offer first due to his own importance but in honor of his tribe.
Or ha'Chayim and Oznayim la'Torah #1: This teaches that he merited to offer first due to his personal level - as he was the first to jump into the Yam Suf 2 - as opposed to the other princes, whose only attribute was being Nasi of a Sheivet.
Ha'amek Davar #1: We find that the greatest Tana'im 3 were not called 'Raban'. Similarly, the greatest Nasi was not called Nasi. 4
Ha'amek Davar #2: The other princes showed their Nesi'us and authority on their day of Simchah, whereas Nachshon had only the Simchah of Mitzvah on his day, for he was sad due to the death of his sister's sons (Nadav and Avihu).
Moshav Zekeinim (citing the Rosh): Because the tribe of Yehudah learned to belittle themselves 5 from Yehudah - when he said to Yosef "Yeishev Na Avd'cha Tachas ha'Na'ar Eved la'Adoni". 6
See also Oznayim la'Torah, DH 'Vay'hi ha'Makriv' #2.
For example, the Zugos.
We find however, that he was called "Nasi" earlier - in 2:3, and later, in 34:19-28 in connection with the division of Eretz Yisrael, the Torah refers to all the princes as Nasi" with the exception of the first three (Yehudah, Shimon, and Binyamin). The Emes le'Yakov there gives a reason for this (PF).
About David ha'Melech too, the Navi writes in Shmuel 1. 17:14 "ve'David hu ha'Katan".
See Vayigash Bereishis, 44:33.
What is the significance of the 'Hey' in ba'Yom ha'Rishon"?
Rashi: Refer to Vayikra 9:1:3:1*.
What are the connotations of the words "le'Mateh Yehudah"?
Rashi: The Torah is merely being Meyaches him to his tribe, 1 but not that he claimed the money for his Korbanos from the members of his tribe. 2
Refer to 7:12:1:2.
Why was Nachshon the first to bring his Korban? Why does the Torah write "Vay'hi ha'Makriv ba'Yom ha'Rishon"?
Da'as Zekeinim and Rosh: It indicates that it was not seemingly correct that he should do so, since Reuven and Shimon were older than him - only Yehudah is always the first. Refer to 2:3:1:1*.
Rosh #2: Nachshon offered first since he was the brother-in-law of Aharon (their wives were sisters).
Oznayim la'Torah (citing the Pesikta): Because Yehudah made himself small when he pleaded with Yosef - his younger brother - to release his younger brother Binyamin, Hashem gave Kavod to his descendent Nachshon to be the first to bring his Korban by the Chanukas ha'Mizbe'ach.
Perhaps we can read this like 'ba'Yam Rishon', even though ba'Yom has a 'Vav', just like the Gemara says in Megilah 25b - to read "Kevodo" like 'Keveido', even though "Kevodo" is written with a 'Vav'! (PF)
