In part one ,first stanza.
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I suppose it depends how much you want to read into this. I tend to think it just means that the Mariner has a special story to tell. It is a magical story of sorts so perhaps there is something left in the Mariner of his albatross killing odyssey. I think retelling this story has a magic of its own hence the glittering eye; it sort of attracts the guest to hear his tale.
Mariners on sailing ships spent many long hours on lookout. In the days before sunglasses their corneas developed cataracts and became hardened by exposure to sun and wind, and took on a glittering aspect as they reflected the light.
you can still see the effect in people with cataracts today.