The consensus of dictionaries and etymologies indicates that μαρμαίρω, "sparkle," is a reduplicated form of a hypothetical μαίρω, meaning "flash." The reduplication may reflect the repeated flashing which produces the sparkling. In fact, the twinkling Dog-star Sirius was sometimes called Μαίρη (LSJ). Then comes the well-known connection to μάρμαρος, at first any sparkling stone, and later marble. Euphony may have required the iota to drop from the noun; if there's a more complicated reason, I'm not familiar with it.
In any case, it sounds like you're blazing some fascinating, if highly hypothetical, trails. I can't substantiate any of your connections; someone wiser than I will have to tell you whether your research will ultimately bear fruit. In the meantime, enjoy the journey!