I just don't see it, both from a value standpoint, and also from looking at Barron himself.
Value - This draft is widely thought of to be weaker than previous drafts, with less difference-makers than in previous drafts. Now, the last time I can remember a draft this weak is 2007, when the Patriots traded their second 1st rounder to SF for their 1st rounder in the 2008's draft. They pretty much stated afterwords (I don't remember who, but someone with knowledge of the draft) that they thought it was a weak class, and that the 28th pick wasn't going to get them a first-round talent. I look for them to do something similar this year, especially as we have a dearth of latter round picks, both this year and next year. Something like 27/31 for a first and a third next year, and a sixth this year.
Now, it's entirely possible that you take the opposite view: that a dearth of talent means you should trade up and grab one of the few blue-chippers available. However, that doesn't make sense economically, as when things are scarce, they increase in value (hence the dramatic rise of Barron, admittedly a very good player). BB, with a background in economics, would know the basic principle of success in economics: buy low, sell high. The market right now just doesn't present that opportunity.
Need - Let me preface this by saying that I think Barron is a very good player. Had he come out in last year's draft, I would have placed him somewhere around #20, about the same value as Prince Amukamara. However, the Patriots, while short on safeties, need a FS. In fact, I believe that as the game shifts to more of a spread-offence, shotgun, passing-oriented attack, that both safety positions will soon be manned by FS types. (Unless you have an Ed Reed-type that can cover the whole field, but let's face it, there's only one Ed Reed.)
That is not the core strength of Barron - his main responsibility, while at Alabama, was to make the last tackle, and be responsible for lining up the secondary. He's certainly an above-average cover man, but 13 interceptions are both a good and bad thing: it shows that QBs weren't afraid to throw in his general vicinity. Whether that's the QBs' mistakes, we don't know, but we do know that Barron doesn't have elite range, and doesn't have the fluidity and ability to move those hips and swivel on a dime that a world-class FS needs.
TL; DR: His lack of elite coverage skills and the high cost needed to acquire him make it very unlikely that Mark Barron will be wearing the Flying Elvis anytime soon.