The correct option is
A that she have
In the given sentence, an action has to be given to the noun 'operation'. Thus, the answer must have a verb. But, a verb 'recommended' is already present in the sentence. Thus, to allow a second verb to be in the sentence, the answer must have a conjunction. Also, the answer must have a pronoun in nominative case that refers to the object 'Elizabeth'.
Option A is correct. In 'that she have', 'that' is a conjunction, 'she' is a nominative pronoun and 'have' is the verb. Thus, it meets all the required conditions and is the correct answer.
Option C is incorrect. 'her have' has no conjunction and the pronoun is in accusative case. Thus, it is incorrect.
Option B is incorrect. 'she has' has no conjunction. Without the conjunction a single clause will have two verbs and the sentence will become grammatically incorrect.
Option D is incorrect. The verb in the first clause is 'recommended' which indicates that the doctor is advising Elizabeth. Introducing 'will have' in the sentence will indicate that it is confirmed that Elizabeth is going to have an operation. Due to this clash of meanings, option D is incorrect.