We can offer another reason why Purim is named after Haman's lots.
The word מגילה is associated with bitachon, as it states, (Tehillim 22:9), יפלטהו 'ה אל גול, "Roll [your problems] onto Hashem, and He will save you." מגילה comes from the word גול, roll, associated with bitachon. And bitachon saved us in the era of Mordechai and Esther.
The Gemara (Megillah 15) asks, "Why did Esther invite Haman to the party...? So the Yidden won't say, 'We have a sister in the palace.'"
Initially, the Yidden trusted that Esther would help them. When Haman made his evil decree, they said, "Esther will certainly speak with Achashveirosh and annul the decree." But then Esther invited Haman to her party. People lost trust in Esther helping them and placed their trust in Hashem. That's when they were saved.
Similarly, when Haman drew lots, he thought he could destroy the Jewish nation, but the lots caused the Jewish nation to turn to Hashem and trust in Him. They felt desperate and turned their eyes to Hashem, and whenever Yidden turn to Hashem, they are saved. Therefore, the yom tov is called פורים, named after the פור. The פור was a positive moment for the Jewish nation because it brought them to place their trust in Hashem.
The Gemara (Megillah 13) states, "When Haman's goral fell on Adar, he was delighted. He said, 'The lot fell on the month that Moshe Rabbeinu died.'"
The Maharal (Or Chadosh) explains, "Everything has a limit as to how long it will endure. Haman drew lots to discover the date the Jewish nation would cease to exist, and it fell on the month that Moshe Rabbeinu was niftar. He thought that the last month of the year, Adar denotes, chas veshalom, the end of the Jewish people."
What was Haman's mistake?
The Maharal explains that Haman didn't know that when a Yid has troubles and feels that there is no hope, he places his trust in Hashem. And when a Yid trusts in Hashem he is immediately helped. We can now understand why Purim is named after the פור. The goral was a positive moment for the Jewish nation because the tragedy caused them to turn to Hashem. And when Yidden turn to Hashem, they are saved.
The Gemara (Megillah 15:) states, "When Esther came to the king's courtyard, the Shechinah left her. She said (Tehillim 22:2), לי-א עזבתני למה לי-א, 'Hashem, Hashem, why did You abandon me?' Esther said: Perhaps the Shechinah left me because I called Achashveirosh a dog, as she said (Tehillim ibid. 22:21), יחידתי כלב מיד נפשי מחרב הצילה, 'Save my soul from the dog....' She immediately corrected this and called Achashveirosh a lion, as she said (ibid.22:22), אריה מפי הושיעני, 'Save me from the mouth of the lion.'"
Why did the Shechinah leave her when she called Achashveirosh a dog?
The Maharsha explains that when one is attacked by a lion, he knows that he needs Hashem to save him. He knows that he can't battle a lion on his own. But when one is attacked by a dog, he feels he can grab a stick and hit the dog, or run away, etc. He doesn't feel completely dependent on Hashem. When Esther called Achashveirosh a dog, this meant she wasn't totally reliant on Hashem. She thought she might be able to manage on her own. This is the reason the Shechinah left her. She immediately corrected herself and called King Achashveirosh a lion. She recognized that she wouldn't be able to succeed without Hashem's help. And when she and the nation trusted in Hashem, the miracles occurred.
