What was their argument? Esther thought that there was no hope of achieving anything by natural means. If Mordechai wanted her to rely on miracles, why risk her life? Miracles could happen without any input on her part. Mordechai disagreed. His message was that a person must do all that he can and rely on Hashem to do the rest. An earlier example of this was at the Red Sea. The miracle of the sea splitting only happened after Nachshon ben Aminadav jumped into the water until it reached his neck.
The people of Papunai were asking Rav Masnah where in the Torah do we find support for Mordechai’s opinion that a person must do all he can and rely on Hashem to do the rest, for the story of Nachshon ben Aminadav was before matan Torah. Rav Masnah answered that we learn this principle from mar dror, from the anointing spices. For Chazal say in Horayos (11:2), “The anointing oil that Moshe made in the desert required many miracles from beginning to end. At the beginning it was only twelve logs. Consider how much the pot absorbs, how much the roots absorb, and how much the fire burns. Yet with this [oil] was anointed the Mishkon and its vessels and Aharon and his sons all seven days of the dedication. And with this were anointed the kohanim gedolim and kings… and that oil lasts until the end of days.”
We find the exact same principle in the story where Elisha producing miraculous oil to save the sons of the wife of a prophet from being sold as slaves to a creditor. Before starting the miracle, Elisha first asked her if she had a small amount of oil. Only then did he pour it into a number of other containers that she borrowed from neighbors. The Zohar (Lech Lecha 88a) writes that Hashem does not produce miracles from nothing. The miracle only started after she had some oil to begin with.
Interestingly, the Yalkut describes the amount of oil Elisha produced in terms very similar to what we saw in the Mishnah of Horayos. Here too, the Yalkut says, there was enough oil produced to pay her creditors and for her sons to live on until techiyas hameisim.
In conclusion, Mordechai’s name was in remembrance of the miraculous anointing oil that would never have sufficed according to the natural order of things. This hinted at Mordechai’s insistence that even if something seems impossible, a person must do everything he can and Hashem will miraculously do the rest.