Which sentence is grammatically correct?

The sentence "He did well on his diet" is grammatically correct because it uses the proper past tense form of the verb "do," which is "did." Additionally, it correctly uses "well" as an adverb to modify the verb "did." In this context, "well" conveys the idea that he adhered successfully to his diet, which is the intended meaning.

In contrast, the other options either misuse verb forms or mix up the adjectives and adverbs. The phrase "He done well" uses "done," which is the past participle form of "do," but it should be paired with the auxiliary verb "has" or "have" to form the present perfect tense. Similarly, "good" is an adjective and should not be used to modify the verb "did," as the adverb "well" is needed to describe how he performed on his diet.