Many judicial decisions are based on subjective opinions, applied to diverse, protected class issues, in the context of the law. For example, deciding what constitutes an "undue burden" for an entire class of Alaskans. If one justice on the court is also a member of that class, he or she may more deeply, internally, and accurately understand what an undue burden truly is, how it feels, and how it impacts one's future. And could help inform the other justices. 
Diversity among decision-makers is critical and gaining acclaim because it discourages "groupthink," and brings improved decision-making. 
In Alaska, women comprise about 47% and men 52% of the population. Of the five AK Supreme Court justices, four are white men and the fifth, Dana Fabe, a woman, is retiring. It is up to Gov. Walker to choose one nominee. And that nominee will certainly be either a highly qualified white man or a highly qualified white woman. But only the white woman will give us a protected-class-diverse court.