The BG used computers to generate a breeding program. This programs main goal was to create the pinnacle of humanity: a Male Wizard - the Kwisatz Haderach.
He would be Bene Gesserit Reverend Mother, Mentat, and Guild Navigator, all in one being. - and just as or more importantly - under the control of the BG.
IIRC, Jessica wasn't "in the know" about the breeding program when she made her mistake. she didn't know what was at stake - a thousand years of careful planning and the ultimate goal in her daughter and then her grand-son. She thought, as you, what's wrong with a son? I can always have a daughter later.. and it's for the man I love.
The "crime" was what caused the BG to lose control of not only the KH, but also lose control of the breeding program for 3500 years.
Additionally, the program would have sealed a thousand year blood feud between the Atreides and Harkonnens.
All of this ruined because of love...
The term originated from the Bene Gesserit's prescient
powers, and their inability to venture into a specific region of
prescient knowledge. This region, though mysterious in nature, was
known to be unattainable to females. Specifically, the spice melange
allowed the Bene Gesserit to unlock genetic memory, but only on the
female side. For reasons unknown, they could not see into the male
side, and the very thought of trying was terrifying to them. Further,
melange also exposed the Sisterhood (and others) to a very limited
form of prescience - thoughts, feelings, images into the very near
future, but no more.
Bene Gesserit Attempt to Create the Kwisatz Haderach
The Bene Gesserit desire to uncover this knowledge and its associated
powers drove them to initiate a long-running genetic breeding program
that would yield a male with mental powers capable of bridging space
and time, and that he would be under direct control of the Sisterhood.
The Bene Gesserit knew that, not only would their Kwisatz Haderach
possess Other Memory on both the male and female side, but that he
would be able to predict the future precisely. He would be Bene
Gesserit Reverend Mother, Mentat, and Guild Navigator, all in one
After the emergence of this male - Paul Atreides - the term Kwisatz
Haderach was also understood as meaning "one who can be many places at
once", and became synonymous with Paul. In time it also came to
encompass his sister, Alia Atreides, and his son Leto Atreides II,
since they all had similar abilities.
The initial Bene Gesserit plan was to breed the daughter of Duke Leto
Atreides to a Harkonnen male, which would produce the Kwisatz
Haderach. This was to be the culmination of more than 10,000 years of
careful breeding. This plan would have seen the end of the
centuries-old feud between the Great Houses Atreides and Harkonnen,
and placed a prescient, Bene Gesserit-controlled male on the Golden
Lion Throne, the Kwisatz Haderach. However, because of her love for
Duke Leto, the Lady Jessica disobeyed her fellow Bene Gesserit and
gave birth to a son instead of a daughter, to give him an heir, Paul.
It seemed fairly certain from early in Paul's life that he would in
fact be the Kwisatz Haderach, since he showed an ability to see into
the future. It was when he was fifteen years of age that the Bene
Gesserit sent a Reverend Mother to test Paul's prescience and his
training in the Bene Gesserit ways. This event appeared to be a
significant catalyst for the events that would befall the universe for
the next several thousand years, since Paul's testing with the Gom
Jabbar, as well as the Sisterhood's silent complicity in his father's
death, proved to instill significant negativity in Paul against the
Sisterhood. As a result, when he reached young adulthood, and ascended
to the Golden Lion Throne, he vowed that he would never be under the
control of the Sisterhood. As a result, the Sisterhood lost control of
their breeding program, their Kwisatz Haderach, and the possibility of
placing a Bene Gesserit on the throne.