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Given a confusion matrix, there's all kind of metrics: Accuracy, Precision, Recall/Sensitivity, Specificity.

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But I haven't seen any name for the ratio between the TN (True Negative) and the sum of TN + FN (False Negative): $$ \frac{TN}{TN + FN} $$

Is there a name for it? If not - why isn't it important enough to have a name?

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That would be the Negative predictive value.

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Yes, it is called Negative Predictive Value (NPV) which measures the accuracy of negative predictions. You can read more about it on wikipedia here https://en.wikipedia.org/wiki/Confusion_matrix

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