Based on what we have learned, we can begin to understand why the first Tanna that Rabeinu HaKadosh chose to mention in Torah she’b’al peh is Rabbi Eliezer. Here is the first Mishnah (Berachos 2a): "מאימתי קורין את שמע בערבין, משעה שהכהנים נכנסים לאכול בתרומתן, עד סוף האשמורה הראשונה דברי רבי אליעזר". From when may we recite Shema in the evenings? From the time that kohanim may enter to eat their terumah until the end of the first watch; these are the words of Rabbi Eliezer. Surely, this is not a coincidence. Hence, we will endeavor to explain why the Tanna Rabbi Eliezer specifically was awarded this honor.
I found an answer to this question in the sefer Ohalei Shem (Likutim) authored by the great Rabbi Shem Klingberg, the head of the Beis Din in the holy community of Zaloshitz, hy”d. He relies on the teaching of the Arizal that Rabbi Eliezer was a gilgul of Reuven, the firstborn of the shevatim. As such, the birthright was rightfully his; so, his gilgul was the first to be mentioned in Torah she’b’al peh.
The problem with this answer is that the birthright was confiscated from Reuven. So, why did Rabeinu HaKadosh see fit to return it to him at the beginning of Torah she’b’al peh? I believe we can expand on this answer based on the assertion of the Arizal presented above. The appellation Rabbi Eliezer HaGadol attests to the fact that by rectifying the flaw of Reuven, the birthright that had been confiscated from Reuven was returned to him. This explains very nicely why Rabbi Eliezer was privileged to be the first Tanna mentioned in Torah she’b’al peh; it was a form of birthright he was entitled to as a firstborn after the tikun was accomplished.
Now, we find something truly fascinating. We have learned in the Mishnah (Avos 2, 1): "רבי אומר, איזו היא דרך ישרה שיבור לו האדם, כל שהיא תפארת לעושיה ותפארת לו מן האדם"—Rebbe says: Which is the proper path a man should choose for himself ? Whatever is a credit (“Tiferes”) to himself and earns him the esteem (“Tiferes”) of fellow men. In Likutei Shas, the Arizal explains why Rabeinu HaKadosh chose to begin Torah she’b’al peh specifically with this teaching. It was because Rebbe was a “nitzotz” of Yaakov Avinu the personification of the midah of “Tiferes.” Therefore, he chose a teaching that emphasizes the virtue and importance of “Tiferes.” Now, Rebbe’s official title was Rabbi Yehudah HaNasi. According to the Megaleh Amukos on Vaeschanan (83), the term נשי"א is an acronym for נ'יצוץ ש'ל י'עקב א'בינו—a “nitzotz” of Yaakov Avinu. Thus, it was fitting for Rabeinu HaKadosh to be the one to return the birthright to Yaakov Avinu’s firstborn.
Furthermore, Yaakov Avinu confiscated the birthright from Reuven with the words: "פחז כמים אל תותר"—unstable as water, you shall not take more. In other words, even though you should have been the “bechor,” “you were as unstable as water”; so, you forfeited your status and privilege. However, after his gilgul made amends for his wrongs, the birthright was returned to him. Hence, his gilgul was called Rabbi Eliezer HaGadol.
So, now, Rabeinu HaKadosh—the “nitzotz” of Yaakov Avinu—comes along and returns the birthright to Rabbi Eliezer—the gilgul of Reuven—that he himself had taken away from Reuven by mentioning him first ahead of all the other Tannaim. We can view this as a form of (Kesubos 16a): "הפה שאסר הוא הפה שהתיר"—the same mouth that has prohibited is the mouth that has permitted. Yaakov confiscated the birthright from Reuven, and Rabeinu HaKadosh, his gilgul, returned it to him—after the tikun accomplished by Rabbi Eliezer HaGadol.
To reinforce what we have learned, we will introduce a “remez” presented in Seder HaDOros in the name of Teshuvot Halachos Ketanot. Before he passes away, Yaakov addresses Reuven and says (Bereishis 49, 3): "ראובן בכורי אתה כחי וראשית אוני יתר שאת ויתר עז"—Reuven, you are my firstborn and my strength and my initial vigor.
He points out that the gematria of the words כח"י וראשי"ת אונ"י equals ז"ה רב"י אליעז"ר ב"ן הורקנו"ס plus one. Simply put, these words allude to the fact that Rabbi Eliezer was a gilgul of Reuven. Furthermore, we can imply from this that when Yaakov took the birthright away from Reuven, he informed him back then that in reality: “Reuven, you are my firstborn”—i.e., I am going to return the birthright to you in the future.
In fact, he specifies when that will happen. With the words כח"י וראשי"ת אונ"י which are equivalent to ז"ה רב"י אליעז"ר ב"ן הורקנו"ס, Yaakov informed Reuven that he was destined to reincarnate into Rabbi Eliezer ben Hurkanos, who would rectify his flaws. At that time, the birthright would be returned to him, but for now you are “as unstable as water.” Nevertheless, we see that Yaakov Avinu was true to his word and his midah of “emet.” His “nitzotz,” Rabbi Yehudah HaNasi, returned the birthright to Reuven by mentioning Rabbi Eliezer, Reuven’s gilgul, first in Torah she’b’al peh.