I am from an Ashkenazi background, and follow the custom of not naming a child the same name as a living immediate family member. My father-in-law’s Hebrew name is Yerachmiel. He was given this name to honor his grandmother, who’s name was Rachel. But no one in the family was actually named after her (i.e. the actual name, Rachel). When I have a daughter, B’ezrat Hashem, will it be permissible for me to name her Rachel (after my wife’s great grandmother), or would that name be considered too similar to my father-in-law’s, who was after all, named after the same person?

You have no problem naming a daughter Rachel.