It's the ARMOR PERK!
You know those pink'ish floating clouds - the most basic resource?
That will give you rep with Eido's armor perk.
😇👍💠
Btw, in comparison I'm almost rank 4, and I thought that was high... I've turned in hundreds of potions to Eido.
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You can also make tonics and turn them in to get rep when it's one of the 2 she wants daily. I've almost ranked her up twice, fully, mainly through that method.