Can I omit some nominations? I don't know enough about some of the moves to give an informed opinion. Assuming I can, this is my list:
1)
Flare Blitz

It would give
Wargle something to actually make use of Encourage, and something to differentiate him from all the other Normal/Flying-types. I'm not really sure what calcs would be relevant, so I'll add them in later.

I also support
Groudon, but I really don't think there's any point in giving it to
Flareon, as due to it's shit HP, shit speed and no priority, FB really won't make much of a difference to its usefulness.

I think
Garchomp is also worth a test, but if it proves broken I'd say we should remove it.
2)
Recover/Slack Off/Softboiled/Roost
Snorlax has wanted this move since forever. Having to rely on Rest, especially after its nerf, is crap. With his good Attack and access to Curse, Snorlax is a very different beast to Blissey, but is severely held back by his lack of instant recovery. Not to mention that he is all but
designed for Slack Off.
188/104/216 Snorlax using Slack Off: 50% ;)

Stealth Rock weaknesses are shit.
Ulgamoth is otherwise awesome. Recovery alleviates the problem, but presents moveslot prioritisation issues, so isn't overpowered.
Yanmega; see above.



The
Regis should get it as well, as they sorely need instant recovery.
3) Sucker Punch
TTar; see The Smog
Weavile is pretty much the archtypeal user of this flavour-wise, and he needs a better STAB move than Night Slash and stronger priority than Ice Shard.
Machamp would also be good with this, as others have said.
4)
Trickery
No nominees from me.
5)
Tail Glow

I definitely support
Gyarados, for reasons mentioned in the OP.
I think
Yanmega could be good with it, but I think he's perfectly built for
Butterfly Dance, if that comes up in a future poll, so for that reason, I'm not going to support him for TG.

I also support
Ampharos for the reasons others have mentioned.

I'd also like to suporrt
Raikou, as since he's an Electric-type it suits him fairly well, though with his 115/115 special offense TG might be OP on him; if this is the case, I suggest testing
Nasty Plot, as the SpDef boost from CM doesn't really do anything for Raikou.
6)
Aura Sphere
Alakazam needs this move more than anything else. Forget Stone Edge, Focus Blast is the worst move in the game, and Zam suits AS flavour-wise pretty well.
Yanmega could really do with a way to deal with Heatran, Lucario and the other 4x Bug-resistant Steel-types outside of HP Ground, which can be taken advantage of by Skarmory.
Modest SpecsMega's Aura Sphere vs:
252/0 Neutral Skarmory: 59.9% - 70.7% (HP Ground does... nothing)
Specially Defensive Skarmory: 40.7% - 48.2% (very likely to KO with Air Slash after SR and lefties)
252/252 Calm Heatran: 53.4% - 63.2% (guaranteed 2HKO with SR)
Choice Scarf Heatran = 97% minimum, almost guaranteed OHKO even without SR
252/252 +SpDef Lucario is never OHKO'd, but
nobody runs that on Lucario, and offensive Luke is OHKO'd by a country mile.
Basically, it does everything HP Ground does, but better.

Latias and
Latios could really do with a move to hit Steel-types supereffective. Surf and T-bolt supply similar neutral coverage, but they could really do with a way to hit some of their key resistors for SE damage. Flavour-wise, Aura Sphere has very Psychic connections, so it fits better than Flamethrower/Fire Blast. This could push one or both over the Uber barrier, though it's still worth a test IMO.
Azelf would be an absolutely excellent user of this move, making its Nasty Plot set that bit more viable, since he would no longer have to rely on HP Fighting for coverage, and he gets a power and accuracy boost to boot.

If
Darkrai is found to be Uber without this move, I suggest we give him it. If he's found to be OU in its abscence, I think this would push him over the edge, so it should be avoided.
I'm unsure whether to give it to
Gengar or not, as he's already a powerful OU poke, and AS would be something to differentiate 'Zam from 'Gar. I'm not voting against Gar, but not for him either.
As a side note, another thing we could do that would help a huge number of pokes without making them overpowered, is simply turn Focus Blast into a Fire Blast clone, with 85% accuracy. Currently, of all the 120BP moves, Focus Blast is the worst, since Thunder has more PP, 30% Para rate, and improves to 100% in Rain, while Blizzard does the same in Hail. There's no reason for FB (or Blizzard, outside of Hail increase) to be 70% accurate, so I propose the first thing we do for the changing-a-current-move part of this project is to improve Focus Blast's accuracy. I only say this here as most pokémon who want Aura Sphere currently have to make do with a shitty Focus Blast.
7)
Wood Hammer
No nominations
8)
Nasty Plot
Absol, for reasons mentioned by later persons (this is an edit)
Alakazam, because with his shit defences CM is pretty awful on him, but NP would be massively useful. A set of NP/{Psychic/Psycho Shock}/{Shadow Ball, Dark Pulse}/Aura Sphere would be awesome.
Gengar, since it suits him ridiculously well for flavour and it'd be massivele useful.
Tyranitar I don't think has been mentioned, but I think he could be a
very interesting user of NP; currently, special sets are
viable, but generally only for the surprise factor etc. With Nasty Plot, Timid 252 SpA TTar reaches 578 SpA, while Modest gets him to 634. And as we all know, TTar's special movepool is awesome. This could be very interesting for mixed sets with Sucker Punch as well. And it makes sense given TTar's Dark typing.
A very out-there suggestion with no real flavour base is
Kingdra; currently, Kingdra's physical sets are more or less better at everything than his special sets. But Nasty Plot could change this; ChestoDD would still be exclusively physical, but Double Dance would be outclassed by NP+RD, as in Rain Kingdra doesn't need the speed boost from DD at all. This would allow Kingdra to be more unpredictable, as currently, if Kingdra comes out, 9 times out of 10 it's DD/Waterfall/Outrage/{Rest/Substitute/Rain Dance}, much of the time the final slot being given away by the hold item. Kingdra maybe fits Tail Glow
slightly better flavour-wise, but tbh neither really fit, this is purely for competitive reasons, and I think TG could be too overpowered, so Nasty Plot it is.
Raikou, if TG is overpowered.

Latios (and to a slightly lesser extent
Latias) would love this move, but I suspect it may push him over the line into Ubers, especially with Aura Sphere as well. Still, it might be interesting to test.
9)
Close Combat
Dragonite; see The Smog.
Weavile; Weavile really needs some love. Currently all his moves are totally underpowered, and since he's paper thin anyway, the defense drops don't really hurt him.
Absol; somebody else gave reasons for this, which I support.
10)
Drill Liner
No nominations.
I might add some more later.