I have an Eye of Ugin and a Cavern of Souls in play, and I'm about to cast an Eldrazi Mimic which I think my opponent might counter.

Can I pay 1 colored mana from Cavern even though Eye has reduced the cost of my Mimic to 0?

No. There is no mana cost to pay.

Thanks for the quick response! I had a feeling that would be the case...