The Warrior Job quests are all given to you by a Roegadyn named Curious Gorge.
The leader of the millers in the "Poor Maid's" FATE chain in La Noscea is named Rabid Ratata.
One of the Elite Marks in Western La Noscea is a crab named Dark Helmet.
The first boss of the Keeper Of The Lake is named Einhänder. And if you know the game that references, then you'll have a good idea of his main gimmick - namely, that he will switch out weapons that he carries underneath, from a bar to a minigun to a cannon.
Quest and Guildleve Titles
A mission that you need to take to obtain a permit to get a Chocobo mount is called My Little Chocobo.
A few quests later and you are sent by the Order of the Twin Adder to find the Sylph elder in a quest called Spirited Away.
There is a quest called Wretched Hive of Villainy. The questgiver opens by saying the location (a "hive" of the bandits known as the Redbelly Wasps) wasn't always a "wretched nest of scum and villainy".
There's even a shout out to This Very Wiki! One of the quests is called "Enemy Mine" and it involves someone in the South Shroud asking the Coeurlclaw Poachers and Redbelly Wasps to look out for Garleans.
The quest to unlock Hard Mode for Tam-Tara Deepcroft is called Corpse Groom.
One guildleve quest in Coerthas is called "We Can Dismember It For You Wholesale", a reference to a Philip K Dick short story titled "We Can Remember It For You Wholesale", which in turn was the basis for the movie Total Recall (1990).
There is a quest in Ul'dah is entitled "Tougher than Leather", the name of two albums by Run-D.M.C. and Willie Nelson, as well as lyrics from the song "Unknown from M.E." from Sonic Adventure.
Of course, there's always the quest given in Coerthas called Dead is Dead.
One vanity pet you can own is a gray cube with legs and wings called the "Demon Brick". It is believed to be a shout-out to World of Warcraft developer Ghostcrawler, who famously expressed his annoyance with theorycrafters by saying they'd use a featureless gray cube for a pet if it gave them a little extra DPS.
This has been debunked - the Art of Eorzea artbook explains that it is actually meant to be a minionized version of the Demon Wall boss.
One of the attacks used in the final fight for Tam-tara Deepcroft Hard Mode is Red Wedding.
The final boss of Hullbreaker Isle is a Kraken, who uses an attack called At World's End.
Continues on into Sastasha (Hard), where the Kraken is summoned by a pirate (who thanks to a mutation resembles Davy Jones) shouting "Release the Kraken!" and a new attack from the Kraken being called "On Stranger Tides"