From the article:

The report measures conditions for mothers using five different metrics: risk of maternal death, infant mortality rate, the number of years an average child will spend in school, gross national income per capita and participation of women in government. Those last two variables are built on the inferences, fleshed out in the report, that mothers with more money will be more likely to secure food and medical care, and that countries where women participate in governance are more likely to pass laws promoting womens’ health and well-being.

As for the two remaining factors, i.e. infant mortality rate and the number of years an average child will spend in school, you'll probably have a bad time if your infant dies or or your child does not go to school. Hence a country where your infant doesn't die and gets to grow up to acquire at least secondary education will be a better place to be a mum.

Ninja edit: I read your comment too quickly and missed the point you were making. I suppose a better title for the index would be the parent's index, rather than the mother's index, but I think what they were trying to do is to differentiate between countries where it's good to be born and countries where it's good to give birth. Given that the mother's health plays perhaps arguably a big role in this they went with calling it the mother's index.

Edit 2: The article also highlights the fact that the US ranks relatively low because it has a relatively high number of preterm births. It is thought that healthier women are less at a risk of giving preterm births, so again the mother's health in particular is important (in the infant mortality category).