Namelessness can have two purposes. Sometimes, to leave a character nameless means that they are not important. In the context of this story, however, we know that the Millers' child is certainly important to his mother and father—William notes that his wife and child are the two people he loves most in the world, while Janet adjures her husband to be careful in operating by thinking of the sickly child as if it were his own. She asks him to think "what it would mean to lose him." So, this child, while nameless, is important to his parents like any other: within the context of their little world, he is very much loved.
In the wider context, however, his namelessness takes on a new meaning. We could interpret it in two ways:
1. To white people, all black people (in the context of this novel) are unimportant, unworthy of being named. This does not seem to be the overriding meaning of the story, however, because there are many other named black people in this novel, and they are allowed by the narrative to tell their own stories.
2. Namelessness connotes a kind of universal applicability. If the child is nameless, he need not be a specific child, and his story does not have to be particular to him. This can therefore suit two purposes. A nameless child's plight can be seen to reflect the plight of many similar children. Additionally, we can infer that, for many white people, black people can seem like a monolith, viewed as a group rather than as individuals.
Compare the Millers' child to the named child, Theodore. Jane thinks that if that white child had been black, his mole would have definitely connoted bad luck; because he is white, she is not certain of this, although she takes precautions anyway. This child, because he is white, will automatically be luckier within the context of the novel than a black child. At the same time, he is given a name at the very moment of his birth, whereas the Millers' son is not named despite being old enough to speak.
There are certainly many unfortunate similarities between the world depicted in this novel and our current world, and these are a result of slavery's legacy. White people can still have a tendency to view black people as a group, rather than as individuals, and it is still the case that white children are born with more innate advantages because of institutional privilege.