The correct answer is: E. No error.
The sentence is grammatically correct. The conjunction “but” is used to join two independent clauses, “I offered him part-time work” and “he turned it over.” The phrase “saying that he would” is a dependent clause that modifies the verb “turned.” The adverb “rather” is used to express preference. The verb “wait” is in the infinitive form, which is used after the word “rather.”
The sentence could be improved by adding more detail. For example, we could say: “I offered him part-time work at the store, but he turned it over, saying that he would rather wait for a full-time job.” This would provide more information about the job that was offered and the reason why the person turned it down.