The correct answer is A. will have been.
The sentence is in the future perfect tense, which is used to talk about an event that will have happened by a specific time in the future. In this case, the event is “he staying at my uncle’s house for three weeks.” The time in the future is “by next Monday.”
The verb “stay” is conjugated in the past participle form, “stayed.” This is because the future perfect tense is formed with the auxiliary verb “will have” and the past participle of the main verb.
The other options are incorrect because they do not use the future perfect tense. Option B, “shall have,” is the past subjunctive form of the verb “to have.” Option C, “will have been,” is the future perfect progressive tense. Option D, “shall have been,” is the past perfect progressive tense.
The future perfect tense is used to talk about an event that will have happened by a specific time in the future. The past subjunctive form is used to talk about something that is not possible or likely. The future perfect progressive tense is used to talk about an event that will have been happening for a period of time by a specific time in the future. The past perfect progressive tense is used to talk about an event that had been happening for a period of time before a specific time in the past.