The correct answer is option (C):
Between him and me
The sentence is "Between he and I, this is a secret." The underlined part is "he and I". We need to determine the correct pronoun case to use after the preposition "between".
Prepositions like "between" govern the objective case of pronouns. This means that the pronouns following "between" should be in their objective form.
Let's look at the pronouns in question:
"He" is a subjective pronoun. Its objective form is "him".
"I" is a subjective pronoun. Its objective form is "me".
Therefore, the correct objective pronouns to use after "between" are "him" and "me". When listing multiple pronouns after a preposition, it is customary to list the objective pronouns in a specific order. While both "him and me" and "me and him" are grammatically correct in terms of pronoun case, "him and me" often follows the convention of putting the other person's pronoun before "me" in informal contexts and a more respectful order in formal contexts. However, the primary error here is the use of subjective pronouns instead of objective ones.
Let's examine the options:
1. Between he and me: This option still uses the subjective pronoun "he".
2. Between me and he: This option uses the objective pronoun "me" but keeps the subjective pronoun "he".
3. Between him and me: This option uses the objective pronouns "him" and "me", which is the grammatically correct choice.
4. Between him and !: This option is nonsensical and includes an exclamation mark where a pronoun should be.
Thus, the correct replacement for "he and I" is "him and me" to ensure the correct objective case is used after the preposition "between".
The final answer is $\boxed{Between him and me}$.