Choose the correct pronoun in parentheses.
some take an unusual approach; for (him, them), a time capsule is the answer.

I assume here that the "some" and the pronoun refer to the same person/people: otherwise the task would not be solvable, because we would not have enough information (it could be a "she" too) otherwise.

if you look at the first part of the sentence, you will see "some take" - the verb is in plural. So it means that the  pronoun also has to be in plural: "them".

So the correct answer is "them".