The part of the game that I am probably least familiar with is special teams. Simply never learned the mechanics of that part of the game before and it isn't covered as much in the football analytics community. That being said, here was an early attempt to quantify the value of punting when deciding between punters in the draft/free agency process.
I did a correlation test between a lot of different ways of measuring punt success and compared the stability year over year as well as the predictive power with next year's values in more traditional counting stats. I seem to have found that Mean Win Probability added on punts without returns was the most predictive (though from what I recall, it still wasn't very predictive) and compared that with percent of punts within the 20 yard line. I find that the first metric is probably more informative since it shows the ability to account for game situation and consistency while avoiding the factors of your coverage unit. Percentage of punts within the 20 yard line suffers from some classic biases because not every punt happens at a location on the field where you can pin the ball back in the 20, sometimes the coverage unit makes a mistake on a perfect punt, better offenses don't punt as often and so they only allow the punter to do harder punts in the first place, etc.
In conclusion, I don't think that I would use any of these earlier attempts as the end-all-be-all for deciding on your team's next punter, but it was fun to investigate.