After coming to the Yankees in the famous "Boone Logan Trade", his career took off for the better. In his four years with the Yankees his ERA, H/9, HR/9, BB/9, and K/9 have all been better than his career average and he posted his first 100+ ERA+ seasons between 2010 through 2013. Despite the ire he has gotten from Yankees fans, he has been very valuable. He's accumulated 1.7 fWAR over 175.2 innings since 2010.
Unfortunately, he hasn't been one of the more valuable lefties in that time; he didn't even rank in the top ten lefty relievers by fWAR in a minimum of 170 innings. He ranks 13th behind Matt Thornton, Glen Perkins, Darren Oliver, Jonny Venters, Rex Brothers, Scott Downs, Antonio Bastardo, Eric O'Flaherty, Phil Coke, Javier Lopez, Craig Breslow, and Hisanori Takahashi, however, he ranked third best in K/9 (10.30) out of those players.
Downs, Lopez, O'Flaherty, and Oliver will all be hitting the open market. Matt Thornton could be a free agent, if the Red Sox decide to buy him out. Logan will be the second youngest out of the bunch, and ranks in the middle of the pack in FIP against lefties this season and the second worst over their careers. He certainly won't be the best option on the market, but he's still pretty decent. Out of these four, possibly five, alternatives to Boone Logan, Darren Oliver is Mariano Rivera-old and Eric O'Flaherty will be recovering from Tommy John surgery.
So there will be definitely two better options over Logan; Scott Downs and Javier Lopez. They made $5 million and $4 million this season, respectively, which is more than Logan's $3.1 million. He will get a pay raise in free agency, so the money will all likely be even. If the Yankees want a one-year deal, it might be worth it to look at Downs or Lopez. If they want someone for multiple years, if they think no one in the system is coming up, then it might just be worth re-signing Logan. Cesar Cabral and Francisco Rondon are really the only lefty specialist the Yankees have in the upper levels of the system and neither are a slam dunk right now.
The biggest wild card in the whole situation is that Logan will need surgery in the offseason to remove a bone spur in his pitching elbow. His ligaments are healthy, but surgery is still surgery, and there's no telling how he will pitch after routine surgery. CC Sabathia didn't have an structural damage when he had surgery, and now look at him. Requiring surgery might allow the Yankees to offer him a lower contract, but will it be worth it if he's not the same come 2014?