Selecting only one of several children in a selected household leads to an unequal probability of selection for all children in the population, or sampling universe. If all children in selected households are included in the survey, the probability of a individual child who lives in a selected household being included in the sample is the same as the probability of his/her household being selected. If that child's household is selected, every child in that household is automatically included in the sample of children. However, if there are two eligible children in a selected household, and the survey team chooses only one, then each of those two children will have only 1/2 the probability of being selected as the probability of their household being selected. That is, if their household is selected, then there is an additional stage of sampling where only one child is selected. As a result, a child in a 2-child household has 1/2 the probability of selection as a child who is alone in the household.
Below is a schematic demonstrating that this unequal probability of selection actually results in a sample which is less representative of all children in the population. The left box, arranged in two pairs of columns, represents a hypothetical population of 15 households (assigned letters A through O) with 20 children who are eligible for the survey (numbered 1 through 20). For example, household K has two children, number 11 and number 12. Note that 10 (50%) of the children in this population live with another child eligible for the survey, and 10 (50%) live alone.
The right box shows the sample of households and children in red font after selecting a systematic random sample of eight households, then selecting one child within each selected household. Note that in this sample of eight children, five (63%) of children live alone and only 3 (37%) have another child in the household. This sample is not representative of all children (numbers 1-20) in the population.
But now if we select all children in each selected household (note two children selected in households K, M, and O), the sample is much more representative of the entire population of 20 children; now 45% are alone and 55% have a sibling.
This wouldn't be very important if single children are the same as children with a young sibling, but you can easily think of reasons why this may not be true: perhaps if there are two young children, the older one was weaned earlier than normal; or perhaps in poor households, two young children sharing very limited food and caring resources do not get as much as if each were alone in the household. Moreover, this potential bias becomes greater in poorer populations which may have fewer household resources to share with two young children. This is because poorer populations tend to have higher fertility rates resulting in a greater proportion of households having more than one eligible child.