As we move towards the end of another NFL season, I am going to surface my off-the-top-of-my-head list of aspects of the current rules that annoy me, with some quick suggestions for changes.

1. End of game kneel-down charades
The use of kneel-downs is an old convention (no more, no less) at the end of a game, where the team that is leading, usually with a full collection of downs, adopts what is known as "the victory formation", snaps the ball, and the quarterback kneels with the ball. The defense is expected to not attempt to push up to the quarterback.
The validity of this convention came under discussion in the 2013 season when Greg Schiano, the coach of the Tampa Bay Buccaneers, ordered his defense to push and attack the New York Giants offense at the end of a game, despite the Giants being in "kneel down" mode. The tactic upset the Giants, and left Schiano defending his orders to his team on the grounds that it is legitimate to attack the quarterback, even at the end of the game, since it could result in a fumble and recovery by the defending team.
Schiano's justification for the tactic, based on his claimed success with it at Rutgers, is BS; Rutgers never achieved any significant success with it during his tenure. They forced a grand total of 2 fumbles in 11 years. This is not exactly a high strike rate. I believe that he actually used the tactic as a motivational aid for his team, a reminder and challenge to them that they have to play for the whole 60 minutes.
Lost in the wider debate about Schiano's tactics is the reality that kneel-downs are the lowest risk approach for the offense at that stage of a game, with only a single ball exchange from the center to the quarterback, minimizing the risk of a fumble or other offensive SNAFU. However, they are a charade and a waste of game time. Players are paid to play for 60 minutes. I am not in favour of rules that can be bent to give the leading team carte blanche to run out the clock. At the end of the game, watching the leading team, if operating on offense, kneeling down and using up one or more downs, taking 40 seconds off the clock each time results in the last 2 minutes of many games being a charade.
The difficult issue is to specify a specific rule change that will eliminate kneel-downs by a team that is leading, but which will also differentiate a kneel-down from (say) spiking the ball, a play often used by a trailing team that is either out of timeouts or does not want to use a timeout and needs to stop the clock.
The most obvious modification would be to change timing within the last 2 minutes of the game so that the play clock stops at the end of a play and does not restart until the next snap if there has been no forward progress on the last play from the offense. That would eliminate all of the value of kneel-downs and force the offense to at least make an attempt to create a positive play. It would not eliminate spiking the ball, since that might still be needed to stop the clock if a team is out of timeouts. Spiking the ball would still be regarded as a lower-risk play than trying to run or pass.
A more radical revision would be to stop the clock after all plays in the last 2 minutes and not re-start the clock until the ball is snapped. I suspect however that the latter idea would not be popular with the TV networks, who are already upset at the increased length of games (mainly due to the proliferation of replay timeouts). I can forsee other impacts, with teams being more willing to spend timeouts earlier in games, since the value of timeouts at the end of a game will be greatly eroded.

2. Rework the 1 point Try rules
When a little-known coach named Bill Belichick thinks that a play is a waste of time because of its high success rate, it is time for the NFL to listen. This play option either needs to be eliminated (leaving only the 2 point conversion option) or the rules need to be modified to make it less of an "automatic" for the offense. Moving the ball back another 10-15 yards would help, but there may also need to be other changes to reduce the almost 100 percent success rate. The article referenced above floats a bunch of other ideas.
Personally I like the idea of adopting the process used in Rugby Union, where the scoring team has to attempt to kick for extra points from a point back up the field in line with the spot at which the ball was grounded in the end zone (note that in Rugby, to score a try, you have to ground the ball in the end zone, not merely break the plane of the goal line). If this was adopted, and (for example) a quarterback threw a touchdown to a receiver in the corner of the end zone, the kicker would be forced to kick the extra point from close to the sideline, which makes the target area above the posts a LOT smaller. To make it fair, the rugby rule of requiring all players to stand behind the goal line until the kicker hits the ball would need to be adopted also.

3. Move kickoff back to the previous position
The moving of the kickoff point forward 5 yards has been a success if the objective was to reduce the incidence of kickoff injuries. Unfortunately, the result is that a very high percentage of kickoffs are null plays, a touchback resulting in the receiving team starting from its own 20 yard line. The referees could just have handed the receiving team the ball and avoided wasting everybody's time. At the very least, move the kickoff point back 10 yards.
But here is a radical thought - most of the injuries result from collisions of players moving at high speed in opposite directions. How about lining the kicking team up along the sidelines, 5 each side, in a zone either side of the center line, except for the kicker? That way, the collisions (whatever they might be) would be more lateral, reducing the impact forces.