I think that the Sox are in a great position. Having Drew on the roster in 2014 is a huge plus (whether you like him or not) UNLESS they trade one of their starters not named Ryan Dempster.
The Sox keep Drew on a 1-year deal. He is likely signed for short money (8-11 MM) and we have great depth on the left side of the infield. If he plays terrific, then just qualify him for 2015 and either sign him for another year or take the draft pick. As Moon has pointed out, this may allow the Sox to keep Bogaerts for another season on the back end of his arbitration years when Xander is in his prime. When Bogaerts IS brought up, he will be able to settle into the majors at a less demanding defensive position and that should only help with his hitting.
The Sox let Drew go. Bogaerts and Middlebrooks will be able to showcase their skills from the beginning of 2014 and the Sox will perserve their starting pitching depth (unless a great deal is on the table). The Sox get a top 37 pick in a loaded 2014 MLB draft with the additional slot $$ for (possibly) paying multiple drafted players.
The Sox sign Drew for multiple years. Most unlikely scenario and my least favorite option. BUT, maybe the Sox have decided that Bogarts is just not a SS. So, keep Bogaerts at 3B, have a very good defensive SS next to him for the forseeable future and keep WMB as great depth and (possibly) Mike Napoli's future replacement or a valuable trade chip.
I would prefer option #2 (let him sign elsewhere and take the pick), but I am slowly coming around to option #1 (sign him for one year). I think that he ends up signing with the Sox for more that last year ($9.5 MM), but less than the $14 MM QO. I think that it will be a $10 - $11 MM, 1-year deal. I think that unless WMB bombs in ST or in April AND May, then Bogaerts starts the season in Pawtucket and will be called up right after the Sox get an additional year of team control (June of 2014).