The Mishna in Taanit 15a itself reveals a common structure underlying each invocation: every figure named — Avraham at Har HaMoriah, Yehoshua at Gilgal, Shmuel at Mitzpah, Eliyahu on Har HaCarmel, Yonah — was answered by God in a moment of acute collective need, and the community in a public fast invokes that precedent, pleading that the same God who proved Himself responsive then respond again now.
Rav Kook's Ein Aya (Berakhot 4:2) provides a key to why Avraham heads the list: he describes Avraham as the foremost of believers, who withstood ten wondrous trials, contending against every force that rose to scatter and dislodge him from the path toward divine light, and who thereby established a firm and enduring standing — making his mode of prayer, the Amidah, the paradigm of steadfast petition.
The uniqueness of the Akeidah as a perpetual source of merit is captured by Rashi (Bereishit 22:14), who cites the Midrash Tanchuma to the effect that God sees the Binding of Isaac every year to forgive Israel and save them from punishment, so that in every future generation it is as if the ashes of Yitzchak are heaped up and serving as atonement — meaning the event at Har HaMoriah was not merely historical but constitutively ongoing.
Bereishit Rabbah (39:11) illuminates the extraordinary spiritual power that made Avraham's prayer efficacious: Rabbi Levi teaches that Avraham would pray for barren women and they were remembered, for the sick and they recovered, and Rav Huna adds that even a sick person who merely saw Avraham would feel relief, while Rabbi Chanina extends this to ships at sea being saved in his merit — grounding the Mishna's invocation in a documented, even cosmic, responsiveness to his intercession.
The Sfat Emet (Bereishit, Vayera 25:4) draws out the generational implication most directly: Avraham is the pillar of chesed, and every punishment that does not befall Israel comes about with his assent; the prayer he activated — if there are ten righteous in a city it is spared — becomes operative guidance for all generations, and in this world, says the Sfat Emet, one must lean upon the attribute of Avraham Avinu.
The Maharal (Gevurot Hashem 29) addresses the nature of the elevation granted to figures such as Avraham, describing it not as the greatness of kings through strength or wealth but as a holy, separated exaltedness — closeness to a divine level — expressed in their mastery over the material world while remaining detached from it, which the Maharal says was the very essence of their greatness.
Ohev Yisrael (Vaetchanan 1) addresses the efficacy of repeated, insistent prayer in this context.