Let's see how this might work.
Some of the rivalries jump off the page as obvious:
Marriages of Convenience
There's some other parings that look workable, even if they don't scream "rivalry."
For these teams, I don't see any pairing or scheme that would be much better than alternatives.
This is where things get dicy.
The current NFL schedule sets up great -- 2 games against the rest of the division, 1 game against one conference division, and one opposite conference division, and a game against the teams that finished in the same place in the other 2 in-conference divisions.
We could just add this game, but sometimes it appears that 16 games it too many. Also, this would result in an odd number of games, meaning some teams would have more home games than away games (and vice versa). There are ways to mitigate this -- have all the rivalry games hosted by the same conference each year so the disparity doesn't unfairly advantage one team in the playoff race, make the extra road game the first tiebreaker, etc. Also, you could have some games at neutral sites -- Eagles-Steelers in Happy Valley, Chiefs/Rams in Colombia, and some of the games would effectively be neutral sites anyway.
Or we could taketh away. The games against the same finishers in other divisions looks like the best target, but that is part of how the league ensures parity, and those are often marquee match-ups (it's why there has been an annual Peyton Manning vs. Patriots game).
Speaking of parity, this presents another problem. Since baseball put a similar scheme in place for interleague rivalries, the Cardinals have been up and the Royals (generally) have been down. An annual home-and-home with the Royals has been a boon to them, especially with their rivals playing tougher competition. A scheme where the 49ers have an annual game with the Raiders while the rest of their division faces tougher competition may not work in a league where every game counts as much as the NFL.
It might be fun to have annual Giants-Jets and Cowboys-Texans games, but it's probably nor worth it if it would also mean an annual Vikings-Bills game and messing up competitive balance.