The correct option is
A since
Answer : C
Explanation:
Adverbs are words that modifies a verb, an adjective, another adverb or a sentence.
Here to correctly complete the sentence we have to fill the blank with an adverb is concerned with time.
The sentence in the question is present perfect tense (has / have + past participle)
Let's understand what the various alternatives mean:
--for : can be used with present perfect to define a particular duration
--when : at what time
--since : is used along with present perfect tense to refer to a point in time
And in the sentence, there is only one point/event in time is marked, i.e. birth of Tom.
Thus clearly the grammatically correct alternative is "since".
Therefore the correct complete sentence is:
He and his wife have lived next to me SINCE their son, Tom, was born.