The Patriots have a clear need to change the RB position in the future. In recent years, they've relied primarily on specialists that can tip plays to the opposing defenses. That strategy isn't necessarily a problem because it can save cap space because a true bellcow RB gets expensive in the free market. However, the one market inefficiency the Patriots can exploit here is getting that bellcow RB in the draft and being proactive with a contract extension. It's not so much of a need that the Patriots offense can't function without one, but rather having someone who can run and catch on the field if the Patriots want to push the tempo in games. In the NFL, predictability is a weakness and if teams finds a weakness they attack it.
What killed the Patriots last year was they didn't have an obvious matchup winner at the RB position after Dion Lewis tore his ACL. I'm not sure how effective he'll be recovering from that injury and if he is still able to make those physics-altering cuts he wowed us with. Either way, the Patriots should relegate him to more of a change of pace back and the primary player on 3rd downs because of his ability to win on those downs. As for RBs, the Patriots only look for players coming out of Pro Style offenses. By this virtue, you can essentially eliminate DJ Foster and Deandre Washington from the list and put Arkansas RB Jonathan Williams in their place.
Kenneth Dixon: Dixon is my favorite RB prospect in the entire draft class. He has the ability to run between the tackles and catch the ball out of the backfield. His pass protection could use some work, but he shows a willingness to improve in that area. The Patriots can initially break in him as the primary early down back and rotate in Lewis/James White as the 3rd down back until Dixon gets more comfortable in this offense. Dixon was initially recruited by Sonny Dykes to play in a spread type offense, but when Dykes got the Cal gig, Skip Holtz replaced him as the coach and changed the offense to suit more of a pro style offense.
Kenyan Drake: Drake is going to be connected to the Patriots heavily in the scouting process due to the type of player Drake is and who his coach at Alabama is. Drake is more in the mold of a Kevin Faulk, who brings a change of pace to the RB position and contributes on Special Teams. The Patriots are more in the need for someone who can grind yards between the tackles, but I won't rule out the possibility that the Patriots target Drake late in the draft and put him up with James White to compete for a roster spot.
Chris Swain: Swain doesn't come from a Pro Style offense, but we've seen Belichick pick up option fullbacks before (Kyle Eckel). The Patriots already do have a fullback on the roster in James Develin, who missed the entire 2015 season with a broken leg. Develin is more than adequate as the team's fullback, although he'll be an unrestricted free agent after the 2016 season so you can argue a case for Swain long term. In the Patriots offense, Swain should only line up as a RB in short-yardage (3rd/4th and 1, goal-line) in a jumbo package. Otherwise, the Patriots will likely use him as a fullback. Of course, there is also the Navy element to his availability although they let LS Joe Cardona play this year after the Patriots drafted him in the 5th round of the 2015 draft. Swain should be a strong Day 3 pickup for whoever takes him.
Jonathan Williams: Williams initially made my list, but I took him off in favor of DJ Foster and now I'm putting him back on the list now that I know about the Patriots preferences at the RB position. Williams missed the 2015 season with a broken foot, taking his only snap on the final kneel-down of the team's season. Williams has run for 2,000 yards in basically two season's worth of carries, so mileage should not be an issue. Williams draft stock should be relatively down because he missed the entire season, so there's a good chance he's still on the board on the 3rd day of the draft for the Patriots. Alex Collins is probably a better fit for the Patriots overall, but Williams could prove to be a better value as a 4th/5th than Collins in the 2nd round.
The Senior Bowl is where these players perform in front of NFL scouts and coaches in practice. For running backs, that means running the ball inside/outside, pass protection, and catching the ball out of the backfield. Bill Belichick and Nick Caserio are personally attending the Senior Bowl to get a look at prospects that could appeal to them on the 2nd and 3rd day of the draft. Perhaps one of these backs will catch their eye.