I certainly wouldn't argue that "I feel good" should be replaced with "I feel well," but I have forgotten what we used to call the adjective in this type of construction. Adjective predicate? Adjective nominative? Phrasal adjective?
That would be a predicate adjective. Instead of an adverb describing how the subject is doing the verb, the predicative adjective simply describes the subject, thanks to the help of a "linking verb" as this website so cutely puts it:
Note that this phenomenon can only occur with verbs that are sort of 'modifications' of "to be". So:
I feel good = My body is well
Those shoes look good = Those shoes are good based on their looks (as opposed to their smell, perhaps).
Rest easy = Be easy (wowee!)
Hope that helps!