The answer to the question above would be that the sentence in which the underlined pronoun is used correctly is the following one: She cooks better than them.
In the sentence in question, the underlined pronoun is "them". This is a personal pronoun in the objective case because it is used as an object. When used as an object, the pronouns can be either a direct or indirect object or the object of a preposition. In this case, it is the object of the preposition "than".