We all know and love (or hate) the bards that give some distinctive flavour to the endgame at Limsa, but there are a lot of opportunities for the other jobs as well which would be fun to explore. Of course these would vary in complexity and content because some jobs are more obvious contenders for out of combat actions. Feel free to expand or diversify with your own ideas.

Astro (See Morningstar1337's post): Tarot reading. Personally I'd like to see a system where you can add a customised message to your cards and different combinations of cards. If you trigger the ability another player can chose to select 2 or 3 cards in front of you and thus activating the customised fortune which you thought up (maybe make it appear like a little message box above your head to avoid chat spam)

Dancer: Uhm... dancing.
It would be fun if Dancers had a menu where they could browse all the animations available to the dancer and make a chain of animations to the best of their ability. You know, a little custom performance

Mechanist: Fireworks
Just something where you select the explosive and fiery effects of the mechanist's tool kit as you see fit and drop an item which then displays those effects without you actually performing the animations. Of course you are stuck for the duration since there is a lot of potential here for being annoying.

Dragoon: Sink it.
A little game for two dragoons. Each of them can pop the job action on each other and then each get a little square assigned on which they have to select 3 tiny spots. Then they have to use their jump on the opponents square to find (and sink) the 3 locations the opponent has chosen.

Ninja: Puppetmaster.
Let the Ninja use his Mudras to summon assigned minions (among those he owns) on his head for a short time.

Monk: Martial arts.
See dancer but more focused on showing some fighting moves. Preferably without all the hit and light effects they usually have.

Whitemage: He loves me, he loves me not.
Let a white mage offer a flower with a randomised amount of petals to people. They can then chose to pluck the petals and hope for the best.

Paladin: Pray battle
Paladins who chose different gods can do a 'pray of' where randomly is chosen which one is the stronger god (I dont know this sounded better in my head by there is no harm in putting it here anyway for you to scrutinise).

Marauder: Lemon juice marathon.
Marauders can drink a few kegs of lemon juice and after a random amount they pass out. Let marauders do a drinking game about who can drink the most lemon juice.

Dark Knight : Edge Pose.
He goes into an edgy pose and people can click on him for random edgy text lines. I'm kidding btw.

Add your ideas and let me know if I should add references to other threads since I'm sure this topic has already seen tons of input by the community.