After talking about the female genitals for thirty years, I'm aware that many of us have a problem finding a word we love to use.
Sex organ sounds a bit like a musical instrument.
Most mothers say vagina when their daughters ask questions about where babies come from. The man's penis goes inside the woman's vagina, children are told, and from that day on, heterosexual intercourse becomes the universal definition of sex. But the vagina is technically the birth canal. It leaves out the inner and outer labia, and the clitoris which is a woman's primary source of sexual pleasure. My pedantic pudenda.