For my benefit, I'm laying out the heredity rules for this imaginary society. If I make an error and this isn't perfectly haplodiploid, let's just pretend this is how it works in this nearly haplodiploid society...okay?
So, in this society:
All men have no fathers, but do have grandfathers.
Daughters have mothers and fathers and are equally related to both, just like actual humans.
Fathers have only daughters, no sons, but their daughters all have 100% of their genes.
Mothers have daughters and sons, but 100% of their sons genes come from them, while daughters have only 50% of their genes (just like humans).
All son's of a single mother are 50% related, just like humans.
All son's and daughters of a single mother are 25% related from the sister's point of view and 50% related from the son's point of view. Insofar as a sister can expect a brother to carry 25% of her genes (she has an extra set of chromasomes he does not share) whereas the brother can expect his sister to carry 50% of all of his genes (since he has only one set of chromasomes).
All daughters of a single mother and father are 75% related.
Grandfathers, and greatgrandfathers, and (etc.) are 50% more related to grandsons than granddaughters
Now assuming a Nuclear family with Mother, Father, daughters and sons, we could assume the evolution of this dynamic:
Fathers would care nothing for their sons and would want them out of the household so they would not take resources from daughters. But grandfathers would care and so sons might be expected to leave the house to live with grandparents at an early age.
Fathers would cherish their daughters and would want them to reproduce, particularly in producing grandsons, as they would be as closely related to their grandsons as human fathers are to their sons. As such, they would be interested in spending resources on grandsons equal to the resources they spend on their daughters.
Mothers would cherish sons, and want to lavish resources on them, placing them at odds with their husbands who care not at all for sons.
Mothers would care for daughters but to a lesser extent than sons (50% less).
Sons would be interested in finding a girl and having daughters.
Daughters would be interested in caring for their sisters and encouraging their mother to have more daughters. As such, their interests would be strongly aligned with their father in this regard, who wants daughters as well. However, dad wants the girls to move out and start having sons, the daughters are more interested in caring for each other and helping mom have more daughters.
Daughters have little interest in brothers, but still more than dad has in sons, but not nearly as much as granddad is rooting for grandsons.
So in one form of this society, males take wives. When a son is born he is tormented by dad and mom is punished for the impertinence of having a son. Sisters shelter him somewhat but he must flee the house for grandpa's protection. Daughters are doted on by dad. Daughters care for one another and the house hold has many daughters all encouraging mom and dad to get busy and make more sisters.
Dad is encouraging his virgin daughters to have sons as soon as they can. So, in this world, Sons born out of wedlock to virgin daughters are cherished in the home by mother and grandfather. While son's born to married couples are the focus of malice and strife and, at best, apathy, and are soon driven from the home of their parents into the home of their grandfather.
Dad is watching and periodically marrying off his daughters to suitors. Keeping enough daughters to care for each other and keep mom tamed, but not so many that he does not have enough daughters out there making more grandsons and granddaughters.
Mom continues to want to have sons and will do it on the sly if at all possible. Otherwise, she must periodically deal with punishment from husband and daughters when she produces an unwanted (to them) son.
That's just one imagining. The rules above could be used to come up with all sorts of other possibilities.