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For "show good manners," the most fitting phrases are "behave gently," "behave yourself," or "act politely/properly," as they capture the essence of polite, courteous conduct expected in company, especially with young ladies, often found in multiple-choice questions about suitable alternatives for good conduct.
Here's why and some common options:
"Behave gently": Directly implies soft, kind actions, fitting the context of showing deference/politeness.
"Behave yourself" / "Act politely" / "Act properly": Broad terms for good conduct, covering appropriate speech and actions.
"Mind your manners": A very common idiomatic way to say someone needs to be more polite.
"Conduct oneself properly" / "Be on best behavior": More formal ways to express the same idea

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