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