The Manot Levi (footnote 5 in the image) says that they skipped Shaul because he was supposed to takes G-d's revenge on Amalek, but he failed. It skips him to indicate that his task was completed by Mordechai and Esther, not Shaul.
Although I'm not sure if this interpretation is correct, when I first read it I interpreted it to mean that Mordechai was taking the place of Shaul, which is why they referred to him as Ben Kish.