A player casts Snapcaster Mage. He has a Hidden Strings and no other instant or sorcery card in his graveyard when it resolves. He says nothing and proceeds to attack and do some random things, then looks to flash back the Hidden Strings.

Does this come under the first instance under GPE Missed Trigger, or the fourth? In other words, since the player did not have a “choice” to make when Snapcaster Mage's ability triggered, does it mean he did not need to announce it until it had a visual effect on the game, which is when he sought to cast the Hidden Strings?

For the avoidance of doubt, you may assume no USC Cheating offence has been committed.

A triggered ability that requires its controller to choose targets (other than ‘target opponent’), modes, or other choices made when the ability is put onto the stack: The controller must announce those choices before they next pass priority.

Seems pretty clear that the acknowledgement must be made when the trigger goes on the stack, since a target is required. Just because there is only one legal target doesn't mean it can be chosen implicitly and resolve without ever mentioning it to the opponent. The only exception in there is “other than ‘target opponent,’” as there is no allowance for “if there is only one possible legal target for any given ability.”

As Josh says, the only time that a target defaults is if it reads ‘target opponent’ in a two player game. All others have to have their targets announced at the correct time, even if there is only one possible target.

Agreed. In order for that trigger to go on the stack, it requires a target at the time of it being put on the stack.

Think of casting a spell. If the spell requires a target, in order for you to cast it, you must have a legal target for it or you cannot cast it.