As an aside, Yoshi-P requested that Ninja players communicate clearly which abilities/aspects of their job they consider 'core' or 'untouchable', and which are more flexible/less important, so that the job designers can better make adjustments or innovate. In honour of that request, I'll lay out my personal opinions on what is integral to Ninja, and what is less important/where I see room to innovate.
I will be using the 6.08 iteration of Ninja as the baseline for these comments, as 6.08 is what I consider to be the best iteration of 6.0 Ninja thus far.
Core Abilities (Do not change)
Mudras
The seven base mudra combinations Ninja currently has (Fuma Shuriken, Raiton, Katon, Hyoton, Doton, Suiton, Huton) are simply core to the class -- with the exception of Hyoton. I believe it would be folly to to change any aspect of these base mudras aside from potencies.
Huton
Similar to above, Huton is a core element of what makes Ninja, well, Ninja. The skill speed boost means Ninja feels fast, as a good Ninja class should. Maintaining Huton is something that you have always had to do, and always should do.
Trick Attack
The classic one-minute party buff, last of its kind, which has been changed to a personal buff in the most recent patch. I would consider Trick Attack's party buff a core aspect of Ninja on par with the mudra combinations. Ninja has a reputation as the 'support' melee class, and Trick Attack is the only 'support' ability Ninja really has.
Hide
This may seem a strange pick for 'Core', but I think being able to Hide is an important part of the fantasy of being a stealthy Ninja, and it would be sad to lose this ability, even if it is only rarely used. The fact that it resets mudra charges makes it feel great in dungeons, and means Ninja can set-up Huton and still have a full burst in its opener.
Shukuchi
In my opinion, this is one of the best mobility abilities in the game, and Shukuchi is great for fulfilling the Ninja fantasy. Drop a smoke bomb and teleport away.
It Feels Good (Little or no changes needed)
Shade Shift
A strong personal shield with a neat animation that just feels good to use. The cooldown is quite long, however. It would be nice to see the duration slightly reduced, as well as the cooldown, to use it more often.
Kassatsu, Hyosho Ranryu & Goka Mekkyaku
These abilities have great animations and feel awesome to use. The only downside is that there's only two options for enhanced mudras, and Goka Mekkyaku is currently weaker than Hyosho Ranryu on two targets.
Armour Crush
Extending Huton as part of your basic GCD just feels good. The name of the ability is somewhat confusing, but I think a GCD that extends Huton should alway be part of the 123(4) combo for Ninja.
Bhavacakra & Hellfrog Medium
Simple spenders for Ninki that do the job. The animations are cool, and stocking them for burst windows is somewhat satisfying. The only downside is, as with Goka Mekkyaku, that Hellfrog Medium is weaker on two targets than Bhavacakra.
Ten Chi Jin & Meisui
This may be somewhat controversial, but I really like how TCJ operates right now. I think that being unable to move without losing the effect is a nice tradeoff for how strong the ability is. Meisui, too, is a fantastic addition that perfectly synergises with the single-target combo of Ten -> Chi -> Jin to convert the Suiton buff into raw Ninki for more damage. It just feels great to use.
Bunshin & Phantom Kamaitachi
I really like these abilities. The fact that they're on a 90 second cooldown provides a decent amount of variety in-between Trick Attack windows. The shadow on Bunshin's animation is very cool, and Phantom Kamaitachi is great not just for the very cool animation, but also for how flexible it is as a disengage tool if being in melee range is dangerous.
Forked Raiju & Fleeting Raiju
I think that the current iteration of the Raijus is really good. The animations are very cool, and the flexibility of being able to choose between a regular GCD and a gapcloser means you have some time-sensitive flexibility. The coolest thing I've done with Fleeting Raiju is successfully cancelled a cape knockback during Pinax in P4S. It felt so good to pull off. While it would be nice if weaponskills didn't cancel stacks of Raiju Ready, it is not too difficult to work around, and with properly planning is not an issue.
Hollow Nozuchi
I love being able to proc Hollow Nozuchi when I'm AOEing in dungeons -- it just feels great to use, and really satisfying to be rewarded for AOEing with even more damage.
Room for Improvement
Throwing Dagger
Currently, the best way to use this ability is not at all. Ninja already has many disengage tools in Phantom Kaimaitachi and Raiton, and given Throwing Dagger's low potency, it simply feels bad to use. While I don't think it should be removed, as throwing daggers is part of the Ninja fantasy in my opinion, there is room for innovation here. Perhaps an 'Enhanced Throwing Dagger' of some kind, or being able to consume a Raiju stack for increased damage?
Mug
Ignoring the party buff added in 6.1, Mug is a very rogue-like skill, and not a very ninja-like skill. I believe that the +40 Ninki gain is a smooth part of the Ninja's kit, but Mug simply does not satisfy a ninja-like fantasy. I propose that Mug be retained, but at a higher level (30+) that it be upgraded into a different, more ninja-like ability, that still provides +40 Ninki, but has a more interesting animation/name.
Assassinate/Dream within a Dream
While Assassinate has in my opinion the single coolest animation of any Ninja ability, neither of their abilities interact with the rest of Ninja's kit, and merely provide raw damage. I would not be particularly upset if they were removed or changed in favour of adding an ability that more directly interacted with the rest of Ninja's kit.
Aeolian Edge Combo & AOE Combo
I hesitated putting this here, as I think the basic 123 combo is quite boring, but I do really like the animation for Aeolian Edge, and it is my opinion that Ninja is already quite inundated with GCDs it wants to use during burst windows. There is the possibility of adding different combo finishers for additional effects, however, similar to what was done with Kiss of the Wasp/Viper and Shadowfang in the past... I'm not sure what can be done for the AOE combo, as it's only two abilities (much like most other jobs), but I don't hold any particular attachment to them, either. Death Blossom is a very cool animation, however.
Ninki I believe there's room for more abilities that make use of the Ninki gauge. Currently, in single-target, you will only ever spend Ninki on two abilities -- Bunshin, when it's off-cooldown, and Bhavacakra when it's not -- and in AOE, the same, but replacing Bhavacakra with Hellfrog Medium. I believe there's potential for an additional ability (or more!) making use of Ninki. For example, consuming 50 Ninki to boost the power of the next mudra, similar to how Kaiten used to work for Samurai.
Enhanced Mudras
I believe there's room for more enhanced Mudras under Kassatsu. Currently, there's only two -- enhanced Hyoton (Hyosho Ranryu), and enhanced Katon (Goka Mekkyaku). It would be very cool to see an enhanced Huton, or an enhanced Suiton, or the like.
Hyoton
The unfortunate forgotten child of the base mudras, Hyoton is almost never used. The only time I can think of when I have preferred to use it over Raiton would be in Eureka, when I was running for my life. Unfortunately, it did not save me. It would be great to see Hyoton changed somehow to have more impact.