There are a couple players at my local store who regularly game the relaxed rules for declaring triggers at drafts and FNM.

Here are two common examples:

Arnold attacks with three creatures attack into a complicated board state against an opponent with open mana and several available blockers. One of them has Aqueous Form on it. Arnold waits until combat is resolved to declare “oh yeah, scry trigger” and immediately goes to scry 1 with a lot of extra information that he shouldn't have had when the scry trigger was supposed to happen.

Angus attacks a Jace, Architect of Thought (at 3 loyalty) with Eidolon of the Great Revel, and Fanatic of Mogis. Nguyen casts Azorius Charm on Eidolon of the Great Revel, and after the Eidolon trigger is clearly missed, Nguyen animates a Mutavault to trade with Fanatic of Mogis. Angus then thinks for a while before announcing “I guess I'll send the Eidolon trigger at Jace.”

How would you handle this as a player and as a judge? Assume in these scenarios that Arnold and Angus are both highly competent magic players who know exactly what they are doing.

Hey!
This seems like it is quite the problem. Seeing as you claim both are highly competent players that know exactly what they are doing, I would have a serious talk with them.

The JAR states that“ Intentionally and knowingly breaking or letting an opponent break game or tournament rules, or lying. (“Bluffing” about cards opponents can't normally see is permitted). ” It goes on to state that Any player
engaging in the following must be removed from your event and, at the Organizer's discretion, removed from
the venue entirely.

Missing a trigger then placing it on the stack at a more advantageous point is intentionally breaking a game rule.

As a judge, I would make an announcement before the next event reminding all players of this policy and institute it with a no strike policy.

Remember to follow DQ procedure correctly.

Well it seems we are wondering into a very serious talk (investigation) with them, to make sure everything is as bad as you say (make sure they know they aren't allowed to purposely miss triggers in order to get them later, or at all for that matter). If it confirms your suspicions that they are doing this on purpose, then this falls under serious issues in the JAR

Intentionally and knowingly breaking or letting an opponent break game or tournament rules, or
lying. (“Bluffing” about cards opponents can't normally see is permitted).

And right before that we get

Any player engaging in the following must be removed from your event and, at the Organizer's discretion, removed from the venue entirely:

The jar sets a clear course of action in this case, assuming they know this isn't allowed.

Players aren't guaranteed their missed triggers at regular REL; it is up to the judge to determine whether the trigger should be put on the stack. The JAR says “don’t add the ability to the stack if significant decisions having been
made based on the effect not happening!”
I definitely wouldn't add the Eidolon trigger to the stack in your second scenario, for example. The first one is debatable, but I would probably allow it.

If there is a pattern of ‘forgetting’ and ‘remembering’ their triggers at advantageous times, I would consider having a serious talk and removing the player in question from the tournament.

One of the things that you can bring up in your talk with them is how this specific problem would be handled at Comp REL. FNM for more experienced players usually is the first step in playing in the larger circuits. We want to teach players the correct way of doing things from the ground level up. Show them the section on cheating in the IPG. If they wouldn't do this at Comp REL then they shouldn't be doing this at FNM.

Definition
A person breaks a rule defined by the tournament documents, lies to a tournament official, or notices an offense
committed in his or her (or a teammate's) match and does not call attention to it.
Additionally, the offense must meet the following criteria for it to be considered Cheating:
• The player must be attempting to gain advantage from his or her action.
• The player must be aware that he or she is doing something illegal.

I generally agree with the advice given in this thread - if players are doing this intentionally, that is going into Serious Problem territory and they should be removed from the event. If they are not doing it intentionally they should be calling judges when the delay is spotted rather than just sorting out the triggers themselves, and the judge gets to decide whether too much has happened to let the player have the trigger at that point.

One thing I want to disagree with is Mike saying to talk about the Competitive REL policy. Regular REL does have its own tournament policy and we should make efforts not to confuse the IPG and the JAR, as they have slightly different goals and underlying philosophies.

Mike wasn't thinking of enforcing Comp REL; just showing it to them so that they know at higher levels that's not acceptable, and what the penalties are. As he said, some start at FNM then go on to the big tournaments. It would actually be helpful for players to know more of what they're getting into at those higher levels and what is not acceptable.

The problem is that it already isn't acceptable at this level.

Also, preaching the IPG to FNM players can lead to them being confused about Regular REL in the future.