Why do we quote the story of Rav Elazar Ben Azaryah in the Haggadah?

Question

In the Haggadah, following the story of the five sages in Benei Berak, we quote the following Mishna (Brachos 1:5)

אמר רבי אלעזר בן עזריה: הרי אני כבן שבעים שנה, ולא זכיתי שתיאמר יציאת מצרים בלילות, עד שדרשה בן זומא, שנאמר:יח "למען תזכר את יום צאתך מארץ מצרים כל ימי חייך" (דברים טז, ג). "ימי חייך" — הימים; "כל ימי חייך" — הלילות.
וחכמים אומרים: "ימי חייך" — העולם הזה; "כל ימי חייך" — להביא לימות המשיח.‏

We then continue to "Baruch Hamakom", and the discussion of the four sons.

This Mishna seems to be wholly unrelated to the Haggadah, as well as to the passages before and after it, as a simple read of the Mishna shows that it is discussing the nightly obligation to mention the Exodus, not the once-a-year obligation of discussing it at length on the night of the Pesach seder.

What do commentaries to the Haggadah answer to this question? (Sources please)

Answer 1

In the Rambam's version of the Haggadah (found at the end of his הלכות חמץ ומצה) this passage begins אמר להם רבי אלעזר בן עזריה, R' Elazar ben Azariah said to them. According to this version this paragraph appears to be an extension of the previous one, meaning that R' Elazar ben Azariah said this to the sages when their students came to tell them it was time for the morning reading of שמע.

Answer 2

The Artscroll treasury Haggada quotes Rav Shlomo Kluger (18th century) to answer this question:

Yetzias Mitzrayim had miracles of a natural nature (e.g the people Walking by foot out of Egypt to the neighboring desert) and miracles of an unnatural nature (e.g the firstborns dying and G-d passing over the Jewish firstborns).

Rav Yosef says a supernatural miracle indicates the greatness of Hashem, Abaye however argues says if a person was greater he would recognize Hashem in natural day to day occurrences and those are more important.(Shabbos 53b).

This dispute brought down in the Haggada is of similar nature. Ben Zoma who says that we mention Exodos at night which is more of a dark time which leads to recalling Yetzias mitzrayim and the unnatural miracles that happened e.g the death of the firstborns and melting of idols which happened at night. However Chachomim hold that one should remember Yetzias Mitzrayim in the time of Moshiach where it will seem "natural" in comparison to the in-gathering of the exile from all 4 corners of the earth.

So when Rav Elazar Ben Azariah white Beard sprouted unnaturally and he became Nasi at the age of 18, he took this as a sign of the vindication of Ben Zoma and Rav Yosefs opinion that the unnatural miracles should be accorded more attention on the night of Pesach where it is a Mitzva to expound on Yetzias Mitzrayim, hence he quoted his own miracle on Seder night and his Preference of Ben Zomas opinion to the other sages to show how it is more important to speak of the unnatural events of Yetzias Mitzrayim on Pesach.

Answer 3

Abarbanel explains in his commentary to the Haggadah that this Mishnah is brought here in the Haggadah as a proof to the Haggadah's earlier statement of וכל המספר ביציאת מצרים הרי זה משובח. The proof is that if we have to mention the Exodus from Egypt every night then it is certainly proper to discuss it on the very night that is dedicated to it.

ועם היות מאמר ר' אלעזר בן עזריה נאמר באמת על זכרון פרשת ציצית הנה הביאו המגיד לראיה על מצות סיפור יציאת מצרים בליל פסח כדי לבאר גודל מעלת המצוה הזאת ר"ל לספר יציאת מצרים בליל פסח מדברי ר' אלעזר בן עזריה שהשתדל להוכיחו אף בכל לילות השנה וכפי דרשת בן זומא וגם לדעת חכמים שאמרו שגם לימות המשיח נתחייב בזה ומק"ו נוציא מזה שאם בכל הלילות נזכיר יציאת מצרים לזכרון אותו הנס שיותל ויותר ראוי לספרו בליל פסח שהוא הזמן המיוחד אליו ולכן כל המרבה לספר בו הרי זה משובח

Answer 4

I will give an answer from Abrabanel

The first relevance is that R.E.B.A mentions the deliverance from Egypt. See at the end of Parashat Reeh, where this verse writen: Deuteronomy 16, 3:

‏ לֹא תֹאכַל עָלָיו חָמֵץ שִׁבְעַת יָמִים תֹּאכַל עָלָיו מַצּוֹת לֶחֶם עֹנִי כִּי בְחִפָּזוֹן יָצָאתָ מֵאֶרֶץ מִצְרַיִם ,
לְמַעַן תִּזְכֹּר אֶת יוֹם צֵאתְךָ מֵאֶרֶץ מִצְרַיִם כֹּל יְמֵי חַיֶּיךָ‏

The first part of the verse is speaking about Korban Pessach:"Thou shalt eat no leavened bread with it; seven days shalt thou eat unleavened bread therewith, even the bread of affliction; for in haste didst thou come forth out of the land of Egypt;".
But the second part ("that thou mayest remember the day when thou camest forth out of the land of Egypt all the days of thy life.") is, following the cited Mishna about the quotidian vesperal mention of the Deliverance from Egypt in prayer (usually the 3e chapter of Kriat Shema).
It seems, following Rabbi El'azar that the verse is not applicable on the Times of Mashiach (as reported in Yalkut Shim'oni parashat bo). Thanks to our mishna, we understand that this minimization applies not for The Korban Pessach and the Hagada. let's look the Yalkut Shim'oni on Parashat Bo siman 218: ‏ ויאמר משה אל העם זכור וגו' - אין לי אלא שמזכירין יציאת מצרים בימים, בלילות מנין? תלמוד לומר: למען תזכרו את יום צאתך מארץ מצרים כל ימי חייך. ימי חייך – הימים. כל ימי חייך – הלילות, כדברי בן זומא.

וחכמים אומרים: ימי חייך - בעולם הזה. כל ימי חייך - להביא לימות המשיח.

א"ל בן זומא: עתידין ישראל שלא להזכיר יצאית מצרים לעתיד לבא, שנאמר: לכן הנה ימים באים וגו' כי אם חי ה' אשר העלה ואשר הביא, מגיד שאין מזכרין יציאת מצרים לעתיד לבא‏
Ben Zoma told them (to Chachomim). In times to come, Sons of Israel will not mention deliverance from Egypt... (he do not spoke about Pessach).

conclusion

Abrabanel says that the statement of R.E.B.A. indicates that if every nights we are required to mention, a fortiori this night, that is why chachamime says above "כל המרבה לספר ביציאת מצריים הרי זה משובח" Additionally, the Hagada quote the statement of chachamim, because if the simple mention is eternal everydays, a fortiori the special mention of the Agada. (see here).
Possibly, the mention of this Mishna is aimed at demonstrating that the end of this verse doe's not concern Hagada shel Pessach. I agree that it is not an obvious answer (see here I am not sure).
But we may explain following the Ikarim (cited in Hagada Divre Shaul) that there are two sorts of memories from yetsiat Mitsrayim. The one is a basis to ask a next delivrance and the second is only gratitude. According to R.E.B.A, the second kind only will remain. And this is this kind of remembering is the same as the Hagada for Leil Haseder.

Answer 5

The simple answer is that of the Rambam, whose text of the Haggadah says אמר להם, that R' Elazar said that statement in that sitting in Bnei Brak.