What is incumbent upon a woman who swore falsely to avoid her talkative neighbor?

You must seek Allah's forgiveness and repent for intentionally lying in your apology, for truthfulness is required in all matters. An oath that was uttered unintentionally, without the intent to solemnize an oath, is not a 'yamin ghamus' (a perjurious oath). Most likely, no expiation is required for this type of oath.