According to the Talmud (Megillah 16a), the verse describes Haman as having a covered head because he was covered with embarrassment.
The Me’am Loez adds that Haman did not want people to recognize him, so he had his head covered.
The Ben Ish Chai writes that there was a custom then in Persia to wear large-brimmed hats. Haman’s daughter’s waste knocked Haman’s hat off his head, ruining it. The gallows he unintentionally built for himself were calculated exactly to fit him with his hat, in order for him to be recognizable. In order to account for this otherwise extra half ama, his hat was nailed to his head on the gallows in order to show everyone that even big advisers are subject to the king’s decree.
According to Ben Ish Chai, Haman’s uncovered head was a reward for Shimi’s wife saving the righteous Achma’atz and Yonasan from Avshalom (Shmuel 2 17:19) by uncovering her hair. Mordechai, being Shimi’s (and his wife’s) descendant, was rescued from Haman’s evil designs in the merit of his ancestor’s pious act.
In the Talmud’s lengthy exposition on this verse (Megillah 12b-13a), the Rabbis note that the verse seemingly mentions these ancestors of Mordechai out of order, skipping around generations. For example, Shimi was a distant descendant of Kish (Targum Sheini to Esther 2:5), not his son. The Rabbis therefore expound on these names as indicating Mordechai’s characteristics. He was the “son of Yair” in that he brightened (“hey’ir”) the eyes of the Jews to prayer; he was the “son of Shimi” in that his prayers were listened to (“shema”) by H-Shem; he was the “son of Kish” in that he knocked (“hikish”) at the Gates of Mercy. The Ohel Moshe asks the question: should not the fact that his prayers were listened to be more important – thus listed before – than his act of “brightening the eyes” of others to pray? After all, his prayers being answered saved the Jews! He answers that, indeed, as powerful as Mordechai’s prayers were, the combined power of the Jews he roused with his “great cry” (Esther 4:1) led to an unprecedented era of teshuva, return, whose cornerstone is prayer.
But like every great man, he was not without his detractors and controversy. Another opinion in the Talmud there (Rava) states that the tribes would deflect from themselves responsibility for Mordechai’s seemingly causing Jewish existence to be threatened in the Purim story, as we will discuss (iy”H) when we get to it (in Esther 3:6). The Jews blamed Yehudah for King David’s (a member of Yehudah) not killing Shimi ben Geira (Shmuel 2 16:7-13 and Melachim 1 2:9), and they blamed Benyamin for King Shaul’s (a member of Benyamin) not killing Agag, ancestor of Haman. Interestingly, Rav Shlomo HaLevi Alkabetz notes that Shaul is noticeably missing in this list of ancestors (see Shmuel 1 9:1). Possibly, this could be a way to avoid embarrassing Mordechai for this relation, especially in light of Shaul’s embarrassing failure to wipe out Amalek leading to the Purim story. Otherwise, Rav Alkabetz ventures to opine that Mordechai could be a “gilgul,” (“reincarnation”) of King Shaul.