My doctor says I have a nevus in my eye. Will it become a cancer?

Possibly. Some locations of nevi are more likely than others to become cancers tha others. Watch all for changes.
Eye nevus. Your doctor is almost certainly referring to a nevus of the choroid, the pigmented layer in the back of the eye under the retina. Nevi in this location are occasionally seen on routine exam and are usually benign. There is no reliable data on the incidence of these becoming melanoma. So we make a note of these and observe them on each visit to see if changes occur.