How Did Patriarchy Begin?

Simple stuff, in the very old times, who brings the food home, makes the rules.

Female members of tribes also were probably raising child from very young, so were busy, not able to go hunting and bring food, either unable to hire by themselves someone to take care of the children.

Not far from all that, male members of tribes probably used the female as low/zero salary workforce, not far from slavery.

There you go, starting from there, you can just make the societies as patriarchal as you want, just because female are just tagged as unable to do valuable work.