Final answer:
The dilemma faced by the characters in the story 'Marriage is a Private Affair' is how to name their children.
Step-by-step explanation:
The dilemma faced by the characters in the story 'Marriage is a Private Affair' is A) how to name their children. In the story, Nanny and Mrs. Penn discuss potential locations for the wedding and mention that they might have it in the new barn, indicating that they are planning for a future together and considering children.
However, the other options listed in the question (B) how to tell the father of their engagement, (C) whether they should remain in the city, and (D) whether Nene should continue teaching) are not dilemmas specifically mentioned in the story.