Answer: B. them
Step-by-step explanation:
The sentence would then be, "I stayed with them while my mom was at work." "my cousins" are people so it wouldn't be "it" and the person speaking is not included in "my cousins" so it can't be "us" or "we". "They" is a subject pronoun which means that it would be used if this sentence was about the cousins. But this sentence is about the speaker so "them" would be the correct pronoun.