The Meaning of the Prayer
Tefillah
Rav Chaim Yosef Dovid Azulai, the Chida, explains that the custom of calling out “Elokah deMeir, aneini” in times of danger comes from the episode in Avodah Zarah in which Rav Meir saved his sister-in-law. The kabbalist Rav Menachem Azariah of Fano (1548 to 1620) explains the intention behind the strange words: the person praying accepts for himself whatever lofty intentions stood behind Rav Meir’s own prayer to Hashem.
Source: Chida, Pesach Einayim on Avodah Zarah 18a-b; Rav Menachem Azariah of Fano, Asarah Maamaros, Maamar Chikur Din 1:26