Extensions for which Patriot veterans?

The concept of an offseason in the NFL is very different when it comes to casual fans and the folks that work for the teams (either as a player, coach or part of the administration). Players get a break after the season is over and before OTAs, then another before training camp. Coaches and administrative personnel are working -- to some degree -- just about every day of the year. For the teams that have completed their minicamps, it's break time for the on-field work until the end of July, and this is the time when some other business can be taken care of, such as contract extensions for players with expiring deals. In the case of the New England Patriots, there are a few candidates that would make sense for a shiny new contract.