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!
The term "subject complement" is applied to "good".
But since "good" is an adjective that functions as predicative within the predicate "feel good", the term "predicative adjective (adjectival)" is sometimes used. The choice between "predicative agjective" or "predicative adjectival" is determined by whether this element is an adjective or an adjectival phrase (in "I feel unbelievably good": "unbelievably good" is a predicative adjectival).