Which statement is preferable: “I am eternally grateful” or “I am grateful eternally”?
The first one. When an adverb is meant to modify an adjective, another adverb, or a participle, it always precedes it. E.g.:
She is incredibly beautiful (adverb + adjective).
He was driving very fast (adverb + adverb).
He is a well-spoken man (adverb + participle).
Plus, your goal is to express how grateful you are; therefore, it's normal that the word "eternally" will be emphasized, thus placed before "grateful".