The blood type is defined by codominance. There are three alleles: Iᴬ (related to blood type A), Iᴮ (related to blood type B) and i. The allele i is recessive, and a homozygous recessive genotype (ii) will culminate in a O blood type (absence of A and B). The alleles Iᴬ and Iᴮ share a codominance pattern.
Therefore, the father must have an Iᴬ allele (because he is blood type A). But we must find out the other allele that he must have so his offspring can present the blood types O and B.
As the blood type O, from the daughter, means that she has a homozygous recessive genotype - ii - and we know that one of her alleles for blood type must come from her father, we can assume that the father has one i allele. The same goes for the mother. If the daughter is ii, then the mother must have an i allele as well.
Putting the two alleles from the father that we have figured out, we can say that the father's genotype is Iᴬi.
We already know that the mother has one i allele. If the son is from the same mother (which is possible), we can assume that the mother will have an Iᴮ allele for sure. As the son has type B blood, we know that he presents an Iᴮ allele. As the father does not present an Iᴮ allele, it must come from the mother and the allele from the father must be an i.
So we can assume that the genotype of the son is Iᴮi.
We can also say that the genotype of the mother is Iᴮi, as the daughter has an i allele from her and the son has an Iᴮ allele from her.
So the genotype of the four family members are:
Father - Iᴬi
Mother - Iᴮi
Daughter - ii
Son - Iᴮi
This can be shown in the following Punnet square:
The Punnet square shows the possible outcomes from this crossing, and indicates the genotypes related to the daughter and the son.