Since 2000, there have been 4,480 player-seasons of at least 200 plate appearances in Major League Baseball. That's not 4,480 different players, mind you, but rather 4,480 times over the past thirteen years in which a player has made 200 or more trips to the plate. Jason Bay's 2012 disaster was one of those player-seasons.
In just 64 of those seasons — a scant 1.4% of them — a player batted less than .200. Two of those players were Mets. One was Joe McEwing ("Super Joe"), who hit .199 in 214 plate appearances for the Bobby Valentine-managed 2002 Mets. McEwing contributed a .242 on-base percentage and a .296 slugging percentage, though he did play every defensive position except pitcher and catcher, so at least he was versatilely awful.
The other Met on the list is Jason Bay, who hit .165 in 215 plate appearances this season. He did so while playing only left field. That .165 average was the eighth-worst since 2000. Of the aforementioned 4,480 player-seasons of at least 200 plate appearances, only seven — .16% — resulted in a lower batting average than Bay accrued in 2012.
If you're curious, the worst batting average since 2000 belonged to Brandon Wood, who hit .146 in 243 plate appearances with the Angels in 2010. You can see the whole list here.