A. The Lots
The Megillah tells the famous story of Haman casting lots to determine the date of the Jews’ annihilation. In fact, the very name “Purim” commemorates that event (Source 1 & 2).
But, as the Rebbe asks, why did we name the holiday a Persian name which the Megillah itself finds necessary to translate? Why isn’t the holiday called Goralos, or something of that nature? (Source 3 — Ibn Ezra about the meaning of Pur).
B. The Amalekite
The Rebbe first pivots to the ancestor of Haman, the Amalekites. Amalek waged war against the Jews, as Rashi says (Source 4), right after they questioned whether G-d was present among them.
What does that mean? How could they possibly raise such a question right after crossing the sea?
The answer is that they questioned G-d’s presence on a regular basis. Amalek represents the argument that G-d might be involved in the affairs of the world at pivotal moments, but He is not involved in day-to-day matters. Nor does He care about our small and insignificant actions; He is too lofty for that.
Note that this is a ‘religious’ argument. Therefore, the Rebbe says, ignore it completely; that nagging feeling is an excuse and justification for a personal failing. Don’t engage it; ignore it completely.
C. Being Born Jewish
These logical arguments are not relevant to us, because our Jewishness transcends logic. G-d didn’t choose us for a reason, and we didn’t get a choice in the matter at all. So you can’t argue that it doesn’t make sense for G-d to care about us — nothing here makes sense at all. Our connection transcends logic; it is our “lot.”
D. Judaism in Persia
The name of Purim is specifically Persian, because the holiday represents the fact that even under gentile subjugation, our transcendent connection to G-d remains strong. It’s so that the non-Jews too, recognize that no matter the circumstances (See Source 5), a Jew remains committed.