A few that get on my nerves:
Official NFL passer rating
It so overvalues completion percentage that a completion that loses yards in any amount is rated at 79.2. Hitting 10 of 10 and losing 67 yards (or for that matter of 999 yards) is rated the same as hitting 5 of 10 for a gain of 67 yards: 79.2. If the 5-of-10 QB gains only 66 yards he has the lower of the two ratings in spite of doing 13.3 yards per attempt better. A quarter of the league's starting QBs had ratings below 79 this year -- they could have improved their ratings by throwing more passes that lost yards.
This should be an easy fix. People just want the one number as a proxy for performance, which I think why this stat is so popular. It shouldn't be a big deal to change it and give people a better measure.
QB win-loss record. I just saw a major sports web site going on in big headlines about "Tom Brady is 14-4 in the playoffs, a staggering .778..." Of course he's not, the Patriots with him at QB are 14-5 now. Football is a team game, not a "punt, pass and kick" competition minus punting and kicking.
Here's a ballpark estimate of the QB's contribution to team strength: Of the 53 players on a roster say 36 significantly contribute in a game. Credit each of the 22 starters with contributing a "share" of the team's strength, and each of the other 14 a half-share, 29 shares total. Now say the QB is worth four other starters (a lot) so he gets three extra shares. The QB's four shares of 32 is 12.5%. (In contrast, sabermetricians say the pitcher is 37% of baseball.)
Or to be empirically objective, PFR.com did a study on what a starting QB is worth and found the answer to be on average 2.3 points more than a backup QB. With about 22 points scored per game on average, that's about 10.5% of team strength. Football really is a team game.
This one is the worst. Yes QB is the most important position on the field. But you can't even attribute the entire output of the offense to the position. So why also attribute the output of the defense to him as well considering he isn't even on the field with the defense?
This stat is more about the media and people being lazy than wanting a simple number they can use.
Clutch performance history. During the Jets-Colts playoff game, how many times did Chris Collingsworth proclaim, "The Colts have the greatest clutch kicker of all, Adam Vinatieri, who's made more big playoff kicks than anyone and never misses when it counts"?
In the 2004 Super Bowl Vinatieri missed from 31 and 36 before "winning" the game with a clutch last-moment 41-yarder. Sure, he's hit more big playoff field goals than anyone else -- because he's played for the Patriots and Colts for 15 years and been in far more playoff games than any other kicker.
I might add that Adam's famous partner in super-clutchness, Tom Brady, who led the last-minute drive to set up Adam for that Super Bowl-winning field goal, did so a few minutes after throwing a 4th-quarter interception on the Carolina 9-yard line that the Panthers took back for a TD -- a 10-to-14 point swing. And for that matter, Joe Montana, before he led the great last-minute drive that finished in "The Catch" and cemented his reputation as maybe the greatest clucth QB ever, had thrown three interceptions. People only remember the last play.
Those are a few of the stats he tackles in the article. I think he is correct about why the media push the stats and why fans like them. Thankfully there are sites like Advanced NFL Stats and Pro-football-reference that put the effort in trying to better measure what happens on the football field.