One piece of questing content that was almost universally beloved by all the players of FFXI were the Star Onion Brigade (SOB) quests.
The NPCs were memorable, and interacted with each other depending on the quest. The SOB encounters and quests almost alway involved other non-SOB NPCs in some way or another.
The reward items for the quests were also highly valued, because they were both low-level and granted a simple, yet significant bonuses. I still have my Promise Badge to this day.
Despite the title of this topic, I'm not proposing a copy-paste of the SOB quests from FFXI to FFXIV. What I am suggesting is a set of sidequests along a similar theme of the SOB quests.
Most people like children, and like interacting with them (even NPC children). A group of children getting into all sorts of minor trouble in FFXIV may be a welcome distraction from the day-to-day grind of guildleves.
There are many examples in entertainment of children organizing for adventure and/or mischief. The Baker Street Irregulars from Sherlock Holmes, the Lost Boys from Peter Pan, the Goonies, ... and yes, the Star Onion Brigade and Young Griffins from FFXI.
Anything patterned off those, or something completely original, would make a good addition to the game, in my opinion.
PS - Thank you, Phogg of Eorzeapedia for reminding me of these fun quests. This is really your post. I'm just typing it for you.