In English the correct gender-neutral term is "he." Some people don't like this for various reasons and choose to substitute "she" as gender neutral, often in an attempt to combat a perception of male dominance or out of a sense of fairness. "They" is also often used as a neutral form but it is incorrect (ungrammatical) when used to refer to a singular entity.
Regardless of the reasons for the origin of the use, "he" and "him" are correct when the gender is unknown or ambiguous. Other forms, no matter how common, are not good English.