The Gemara (Avodah Zarah 18.) relates that Reb Meir's sister-in-law was imprisoned, and Reb Meir bribed a prison guard to free her. The guard asked, "What should I do if I am caught?" Reb Meir told him, "Say ענני דמאיר אלקא, 'The G-d of Meir, save me!'" The guard said this and was saved. The government caught him, and they hung him at the stake to crucify him, but when he said ענני דמאיר אלקא, miraculously, they took him down. They didn't understand why they let him off. He told them that Reb Meir told him that when he said this, he would be saved.
Chazal (Bereishis Rabba 94:5) say that one mustn't associate Hashem's name with someone alive. We say אלקי אברהם, אלקי יצחק, ואלקי יעקב after their petirah, but when they lived, it wouldn’t be proper to attach Hashem's name to a person who has free will, and his future צדקות isn't yet determined. So how did Reb Meir tell the guard to say דמאיר אלקא, "The G-d of Reb Meir"?
The Maharsha answers that מאיר hints at Reb Meir's name, but it wasn't the primary intention. ענני דמאיר אלקא means, "Hashem Who shines His light of the sun to the entire world should answer my prayers."
The Maharsha's second answer is that ענני דמאיר אלקא means that "You are the G-d Who shone Your light for us in ancient Greece with the miracles of the Chanukah lecht... He should answer my tefillos and save me in miraculous ways to be saved from their hands."
Many people recite the words ענני דמאיר אלקא when they give tzedakah. According to the Maharsha's explanation, we are asking Hashem to perform His miracles for us, as he had by the miracles of Chanukah.
