Final answer:
The correct form of the pronoun should match the singular indefinite antecedent 'no one'; therefore, the sentence should be 'No one managed to find his or her way to the cabin until after dark.' To correct this error, you can change 'their' to 'his or her' or 'their' to 'the'.
Step-by-step explanation:
The sentence 'No one managed to find their way to the cabin until after dark' contains a pronoun-antecedent agreement error. The antecedent 'no one' is singular, and thus the pronoun should also be singular to match in number. The corrected sentence should read: 'No one managed to find his or her way to the cabin until after dark.'
This issue arises frequently with indefinite pronouns such as 'everyone' and 'no one', which despite referring to more than one person, are grammatically considered singular and require singular pronouns for correct agreement. A writer can also rephrase the sentence to avoid gender-specific language if desired: 'No one managed to find the way to the cabin until after dark.', but this might slightly change the original meaning.