This is derived from the famous story about the Roman guard being saved after invoking R' Meir's merit (Avoda Zara 18a,b). The Medrash Talpiyot writes that someone who lost something should promise to donate some money for lamp oil for the merit of R' Meir's soul. The Chida also cites the custom to donate oil or money for the merit of R' Meir's soul when someone is in any sort of distress. He adds that the person would also recite the prayer of the Roman guard: "Elaka d'Rabbi Meir aneini." The Chida cites the Rama miPano that the supplicant's intention should simply be that he accedes to whatever the lofty intentions were behind R' Meir's own prayer to the Almighty.
Beginning over two hundred years ago, charitable organizations, particularly those benefiting the poor and scholars in the Land of Israel, began to spring up under the name of "Rabbi Meir Ba'al Haness." This includes a well known Jerusalem charity founded in 1860 by R' Shmuel Salant. Since then, it has become popular to start charitable organizations under that name. These organizations typically cite the Talmud's story about R' Meir and assure people that they will be saved from trouble or find lost objects if they donate to that organization, possibly with the additional requirement of reciting a formula including the Roman guard's prayer and a Midrashic teaching (B'reishis Rabba, 53:14) about finding hidden things (e.g., see here).
In reality, an organization that decides to name itself something like "Rabbi Meir Baal Haness" does not automatically acquire a superior claim to the segula described in the first paragraph.