For example, suppose that in a troglodyte encounter, a PC fails his save versus the enemy's stench aura, and is nauseated for n rounds. On round n+1, does he need to reroll that save, or is he considered to be used to it by that time?
In Pathfinder, Stench has the universal rule including:
... Creatures that successfully save cannot be affected by the same creature's stench for 24 hours. ...
With nothing supporting that failed saves confers a similar immunity. Frightful presence has a similar rule to stench:
... An opponent that succeeds on the saving throw is immune to that same creature's frightful presence for 24 hours. ...
These are the only aura-type Universal monster rules that describe anything about immunity. Another aura-type ability, Fear Aura, says nothing about immunity. Thus, Fear Aura would only give immunity to itself if the monster's descriptive text explicitly grants it. Similarly, descriptive text in a monster entry may grant immunity to a creature's stench or frightful presence abilities.
Troglodytes have no such descriptive text, so a failed save causes the sickened condition for 10 rounds. Anytime after the 10th round where they are in the aura they must make a new save, where failure is another 10 rounds and success grants immunity for 24 hours.
As the question is tagged [Pathfinder] and [D&D 3.5], I checked the OGL stuff from 3.5 and found that none of the referenced wordings changed (Troglodyte's stench save text is word-for-word, as is the saving throw immunity for Frightful Presence)
In response to the titular question, Pathfinder's universal rules and the 3.5 equivalent rules state that, in general, yes, Aura-based effects require a new save every time they wear off.
Characters who fail their save against an aura ability, such as stench or frightful presence, are affected for the listed duration, regardless of them leaving the aura or not. After the duration elapses, if they are re-exposed to an effect that they have not saved against, they will need to make another saving throw and possibly be affected again. Saving generally renders the character immune to that effect from that creature as these abilities generally have a line in their description that says:
Creatures that successfully save cannot be affected by the same creature's [ability] for 24 hours.
It's worth noting that a creature in range of multiple auras, such as several troglodytes, will need to make a save against each aura. The effects typically won't stack, but it does increase the likelihood of failure.