The Sox' biggest need has been the same all season: they need relief pitching. Theo did a very poor job this off-season in acquiring bullpen help. He relied too much on two pitchers who were awful in the second-half (Ramirez and Delcarmen) and one that has been steadily declining (Oki). Even Paps struggled at times last year. There was no reason to assume this group would be good before the season, and they have sucked as a group pretty much all year.
No AL contending team has a pen nearly as bad as the Sox (unless you consider the Angels a contender). By ERA, Boston's relief corps are the 4th worst in the AL. ERA measures results based on ER, which are subject to the decision of the official scorer. When you look at a metric that measures performance--FIP--the Sox are the worst pen in the AL by a fairly large margin. This group has trouble holding leads. That's an especially big problem right now because Boston's depleted offense is having trouble scoring.
The Sox have only one reliable arm: Bard. Atchison, by the numbers, is the second-best. He has been pretty good and is Theo's only off-season "find." Paps hasn't been very good, and Ramirez, Oki, and MDC have flat-out sucked. MDC has been below replacement level.
The question is what's out there. There are interesting pitchers like Chad Qualls, who has a horrible ERA but a good FIP and K-rate. The problem with acquiring bullpen help is the cost. Teams tend to pay too much for relief pitchers in terms of salary and prospects. Another thing to avoid are pitchers who look good solely based on ERA, which is bad at predicting future performance. Kyle Farnsworth might be available. He has had a good year. Scott Downs is very good and consistent. But as a potential Type-A FA, he'll cost a lot. There are reports that the Sox are interested in Marlins closer Leo Nunez, who has been outstanding this year. The problem with Nunez is cost, and the fact that this is pretty much his only good year. His first half success might be a fluke. But it's also possible that Nunez has reinvented himself. He is throwing his change far more than in the past and his GB-rate and K-rate have increased significantly.
Good relievers are hard to find. It's generally not a good idea to overpay for them. But the Sox have a glaring need. The only way they can win with their current pen would be if every injured player comes back completely healthy and the team's offense regains its form and the starters consistently go deep in games.
Tampa has one of the best bullpens in the AL. That's the only edge they have over a healthy Sox team. The Rays offense and starters are really not as good as people think they are. If healthy, even with a poor pen, the Sox are good enough to make a run. They are better than Tampa. The problem is they aren't healthy, and probably won't be for a few weeks.