The correct option is
D Can
In the given sentence, 'be' is an incomplete verb. Thus, it requires an auxiliary/modal to complete it. Thus, the answer is an auxiliary/modal verb. The sentence asks whether it is possible to go next week. Thus, the subject is asking someone, if they are able to go next week. Thus, the answer must demonstrate this ability to go.
Option A is correct. 'Can' means 'able to'. Thus, it correctly demonstrates the ability of the person being asked, to go. It clearly asks whether the person will be able to go next week and so it is correct.
Option B is incorrect. 'Might' indicates a possibility and not the ability to go. Thus, it asks whether it is a possibility for the person to go or not. It does not convey the correct meaning and thus, it is not the correct answer.
Option C is incorrect. 'Rather' is not an auxiliary/modal verb. Also, 'Rather' cannot be used to ask questions and thus, using it will make the sentence grammatically incorrect.
Option D is incorrect. 'May' indicates permission and not the ability to go. It indicates that the subject is asking for permission to go next week. Thus, it does not covey the correct meaning and is not the correct answer.