Blood group probability question
I think the quoted answer is correct.  

First, the probability of getting a son( and not a daughter) is $P_s=0.5$.
Next, since both the parents are heterozygous, the probability of genotype $hh$ is $P_h=0.25$. This genotype will not produce any antigen precursor, and hence there will be no antigens on RBC. As blood group O is defined by the absence of antigens, this can be classified as blood group O.For the expression of the antigen (dominant at the $H$ locus), the probability is $P_H=0.75$.  

Again owing to the heterozygosity of both the parents at $I$ locus, the probability of getting a blood group with genotype $I^oI^o$ is $P_I=0.25$. This is the probability, that if the antigen precursor is made, $75%$% of the times the expressed blood group will be O. Therefore the probability of blood group O due to homozygous $I$ locus and dominant $H$ locus is $P_I\times P_H=3/16$.  

Net probability for blood group O=(probability due to homozygous h+probability due to homozygous I and dominant H)*probability of a son.

$$P_{net}=P_s(P_h+P_HP_I)\approx 0.22$$