R’ Dovid Feinstein points out that, since Mordechai raised and taught Esther, he is in a sense her father. When Mordechai references Esther’s “father’s house,” he is saying that her apathy to the needs of the Jewish people will be a mark of shame upon him.
Pachad Yitzchak writes that prayer is the tool of our ancestors, so Mordechai is telling Esther to utilize the power of her “father’s house” – prayer – to save the Jews from their current threat. When someone approaches an earthly king, it is one thing to provide him with a gift, but something altogether more powerful if one has the references. The king would be more likely to listen to the request because he feels like he has more of a connection with the requester.
In explaining this verse, R’ Henoch Leibowitz quotes a Midrash (Tehillim 22) that advises people to “push away with the right hand, and pull people in with the left.” In this case, Mordechai’s methods of convincing Esther to approach the king include “pulling with the left” by his reminding her of her noble, royal roots, and also “pushing away with the right” by warning her to not lose her chance. As R’ Leibowitz continues, if Esther – as righteous as she is – needs this form of convincing, how much more-so do we need to utilize this in our relationships with people. Instead of yelling at a child for doing something wrong, it is important to tell the child, “Doing this is beneath you.”
According to the Akeidas Yitzchak, Mordechai’s reference to Esther’s “father’s house” was meant to emphasize that, considering the precarious state of the Jewish people, she should use her Jewish lineage as an explanation as to why she should be allowed to visit the king unbidden.
The Alshich and the Megillas Sesarim both say that the “father’s house” is a reference to King Shaul, and his sin of allowing Agag to live when he had the chance to fulfill the command to obliterate Amalek. It thus become Esther’s duty to undo that error.
R’ Yehonason Eibshutz notes historically, there is always someone standing in the way of the Jews earning their rescue. In this case, it was Haman. Mordechai was thus telling Esther that he could, himself, get rid of Haman, but that would not make up for Esther’s ancestor’s mistake, which only she could accomplish. Halachically, Esther’s going to Achashverosh voluntarily would forbid her to Mordechai as a wife forever.
The Ginzei HaMelech also points out that Shaul did go through the steps of teshuva (Shmuel 1 15:26, 28). This being the case, why does Esther need to fix his error? Although regret is one step in teshuva, the result of his actions still remained. There is a story of a woman who felt her husband was emotionally abusive. The rabbi she consulted told her to purchase a block of wood and bag of nails. Each time she felt abused, he said, she should hammer a nail into the block of wood. After a few such incidents, the husband became curious about the loud knocking his wife would initiate after each fight. He asked her about it, and the wife told him what the rav had said, and showed him this porcupine of a block of wood. He instantly felt regret for his past deeds, and he made a deal that for every nice act of his toward her, she would remove one nail. Eventually, the block was nail-free. The husband said, “Look! It’s all better! There are no more nails!” “Yes,” she said, “The nails are gone…but the holes are still there.” A sin can be erased, but the consequences of that sin can last forever.