By next Monday he . . . . . . . . staying at my uncle's house for three weeks.

The question tests your understanding of future perfect continuous tense.
Future perfect continuous tense is used to describe an action that will be in progress for a period of time before a specific point in the future. The structure is: will/shall + have + been + present participle (verb + -ing).

The sentence requires a verb tense that indicates an action completed over a period of time, ending at a specific point in the future (next Monday).

Let's analyze the options:
Option A: will have - This is the future perfect tense. It indicates the completion of an action by a specific time in the future. It doesn't show the duration of the action. It would be correct in a sentence like: "By next Monday, he will have finished his work." This is incorrect because the sentence indicates the action is ongoing.
Option B: shall have - This is very similar to "will have," but "shall" is less common in modern English, particularly in American English. The same logic as Option A applies here. It’s also grammatically incorrect in this context.
Option C: will have been - This is the correct answer. "Will have been staying" perfectly expresses an action (staying) that will have been in progress for a duration (three weeks) before a specific point in the future (next Monday).
Option D: shall have been - Similar to Option B, this is less common and grammatically incorrect for this sentence. The same logic as Option C and its grammatical limitations apply here.

Therefore, the correct answer is Option C: will have been. The complete sentence is: "By next Monday he will have been staying at my uncle's house for three weeks."