The Order of the Green Hand is/was something that has always fascinated me for some reason. I am not sure there is much to it except worldbuilding (maybe you could tie it to the Isle of Faces/greenseers?). That said, I still thought it would be discussion for a Saturday.

Anything and Everything about the Order of the Green Hand

The Order of the Green Hand was a knightly order founded by House Gardener (original Kings of the Reach and seated at Highgarden). Their sigil was a green hand and they only offered membership to "those both virtuous and skilled in arms".

Background

House Gardener was descended from Garth Greenhand (as are numerous houses in the Reach):

Of all these, the greatest was his firstborn, Garth the Gardener, who made his home on the hill atop the Mander that in time became known as Highgarden, and wore a crown of flowers and vines. All of Garth Greenhand's other children did the Gardener homage as the rightful king of all men, everywhere. From his loins sprang House Gardener, whose kings ruled the Reach beneath the banner of a green hand for many thousands of years, until Aegon the Dragon and his sisters came to Westeros. -TWOIAF, The Reach: Garth Greenhand

and:

The Queen of Thorns snorted. "So can the Florents, the Rowans, the Oakhearts, and half the other noble houses of the south. Garth liked to plant his seed in fertile ground, they say. I shouldn't wonder that more than his hands were green." -ASOS, Sansa I

and:

In his day there were still seven kings in the Seven Kingdoms, and Highgarden and the Rock were oft at war. The green kings ruled us then, the Gardeners. They were of the blood of old Garth Greenhand, and a green hand upon a white field was their kingly banner. Gyles the Third took his banners east, to war against the Storm King, and Wilbert's brothers all went with him**, for in those days the chequy lion always flew beside the green hand when the King of the Reach went forth to battle.** -The Sworn Sword

and:

Though a man of modest gifts, Mern VI had able counsel in his stewards. Ser Osmund Tyrell was succeeded in that office by his son, Ser Robert, and later by a grandson, Lorent. Relying on their acumen, Mern VI ruled well, rebuilding Highgarden and doing much and more to restore House Gardener and the Reach. His son, Garth XI, did the rest, taking such a terrible vengeance upon the Dornishmen that Lord Hightower said afterward that the Red Mountains had been green until Garth painted them with Dornish blood. For the remainder of his long reign, the king was known as Garth the Painter.

And so it went, king after king, in war and peace. Yet through it all the green hand flew proudly across the Reach, until King Mern IX rode out to meet Aegon Targaryen and his sisters upon the Field of Fire. -TWOIAF, The Reach: Andals in the Reach

The Order itself

Invited the "greatest champion, who were pure, honorable and virous as well as skilled at arms":

No seat in the Seven Kingdoms has been more celebrated in song than Highgarden, and small wonder, for the Tyrells and the Gardeners before them have made their court a place of culture and music and high art. In the days before the Conquest, the Kings of the Reach and their queens presided over tourneys of love and beauty, where the greatest knights of the Reach competed for the love of the fairest maids not only with feats of arms, but with song, poetry, and demonstrations of virtue, piety, and chaste devotion. The greatest champions, men as pure and honorable and virtuous as they were skilled at arms, were honored with invitations to join the Order of the Green Hand. -TWOIAF: The Reach: Highgarden

While it doesn't seem like any type of sword was given out, nor was it limited to one person of a single House, there are some mirrors in the Sword of the Morning (not saying the are in any way related):

George said the Sword of the Morning is always a member of House Dayne, someone who is deemed worthy of wielding Dawn as decided within the House, that whoever it is would have to earn the right to wield it. -SSM, US Signing Tour: 29 November 2005

and:

Though many houses have their heirloom swords, they mostly pass the blades down from lord to lord. Some, such as the Corbrays have done, may lend the blade to a son or brother for his lifetime, only to have it return to the lord. But that is not the way of House Dayne. The wielder of Dawn is always given the title of Sword of the Morning, and only a knight of House Dayne who is deemed worthy can carry it.

Destruction of the Order

While they seem like the most chivalrous of the Reach, they were extinguished on the Field of Fire (during Aegon's Invasion):

Though the last members of that noble order perished beside their king on the Field of Fire (save in White Harbor, where the knights of House Manderly still profess membership), their traditions are still remembered in the Reach, where the Tyrells continue to uphold all that is best in knighthood and chivalry. Their Tourney of the Field of Roses in the reign of Jaehaerys I, the Old King, was famed far and wide as the greatest tourney in a generation, and many other great tourneys have been held in the Reach in more recent days. -TWOIAF: The Reach: Highgarden

and:

"The Field of Fire must have looked like this," Ser Eustace said. "It was there our woes began, two hundred years ago. The last of the green kings perished on that field, with the finest flowers of the Reach around him. My father said the dragonfire burned so hot that their swords melted in their hands. Afterward the blades were gathered up, and went to make the Iron Throne. Highgarden passed from kings to stewards, and the Osgreys dwindled and diminished, until the Marshalls of the Northmarch were no more than landed knights bound in fealty to the Rowans." -The Sworn Sword

and:

The Targaryens waited until the two kings sounded their trumpets and started forward beneath a sea of banners. King Mern himself led the charge against the center on his golden stallion, his son Gawen beside him with his banner, a great green hand upon a field of white. Roaring and screaming, urged on by horns and drums, the Gardeners and Lannisters charged through a storm of arrows down onto their foes, sweeping aside the Targaryen spearmen, shattering their ranks. But by then Aegon and his sisters were in the air. -TWOIAF: The Reign of the Dragons: The Conquest

I would assume the Order was part of the "Iron Fist"

Together the two kings commanded the mightiest host ever seen in Westeros: an army fifty-five thousand strong, including some six hundred lords great and small and more than five thousand mounted knights. "Our iron fist," boasted King Mern. His four sons rode beside him, and both of his young grandsons attended him as squires. -TWOIAF, The Reign of the Dragons: The Conquest

If interested: Extinct Houses of Ice and Fire Part I and Part II

Current Status

Knights of House Manderly (originally from the Reach) still claim it the title:

Though the last members of that noble order perished beside their king on the Field of Fire (save in White Harbor, where the knights of House Manderly still profess membership), their traditions are still remembered in the Reach**,** where the Tyrells continue to uphold all that is best in knighthood and chivalry. Their Tourney of the Field of Roses in the reign of Jaehaerys I, the Old King, was famed far and wide as the greatest tourney in a generation, and many other great tourneys have been held in the Reach in more recent days. -TWOIAF: The Reach: Highgarden

Lord Wyman claims it as one of his titles:

None chose to honor Davos with a name. The maester was the first to speak. "You stand before Wyman Manderly, Lord of White Harbor and Warden of the White Knife, Shield of the Faith, Defender of the Dispossessed, Lord Marshal of the Mander, a Knight of the Order of the Green Hand," he said. "In the Merman's Court, it is customary for vassals and petitioners to kneel." -ADWD, Davos III

House Manderly seems to have been chased from the Reach by the Gardeners (and possible the Order of the Green Hand):

"The storms did you a kindness, blowing you to my door," Lord Godric said. "You'd have found a cold welcome in White Harbor. You come too late, ser. Lord Wyman means to bend his knee, and not to Stannis." He took a swallow of his beer. "The Manderlys are no northmen, not down deep. 'Twas no more than nine hundred years ago when they came north, laden down with all their gold and gods. They'd been great lords on the Mander until they overreached themselves and the green hands slapped them down. The wolf king took their gold, but he gave them land and let them keep their gods." -ADWD, Davos I

Houses

Its very possible that these houses' sigil alludes solely to Garth Greenhand, etc. but it also should be noted that not all of them are confirmed descendants of of Garth.

{House Gardener of Highgarden} - A green hand on white

House Chester of Greenshield - A green hand on a gold escutcheon on a green field, a border of red rayonne

House Uffering - Oak and iron paly, beneath a white chief with seven green hands -Westeros.org (Heraldry)

And while these Hands aren't green I will note them as well:

House Oldflowers - Ten white hands on green, 4-3-2-1, beneath a red bend sinister

House Hutcheson - Two red bendlets on gold, a red hand below pointing to a red sun above

Thoughts

Its interesting to note that the Reach Kings/Order of the Green Hand chased the Manderlys from the Reach and they settled in the North, while the Kings of Winter/Wargs possibly chased the Blackwoods from the North and they settled in the Riverlands:

Amongst the houses reduced from royals to vassals we can count the Flints of Breakstone Hill, the Slates of Blackpool, the Umbers of Last Hearth, the Lockes of Oldcastle, the Glovers of Deepwood Motte, the Fishers of the Stony Shore, the Ryders of the Rills...and mayhaps even the Blackwoods of Raventree, whose own family traditions insist they once ruled most of the wolfswood before being driven from their lands by the Kings of Winter (certain runic records support this claim, if Maester Barneby's translations can be trusted). -TWOIAF, The North: The Kings of Winter

Its possible that the Order was established to fight the Ironborn (although this probably alludes to the Gardener Kings):

In ancient days, the ironborn had boldly sailed the river road and plundered all along the Mander and its vassal streams . . . until the kings of the green hand had armed the fisherfolk on the four small islands off the Mander's mouth and named them his shields. -AFFC, The Reaver

If you are interested in comparing the Order to the Green men, etc. this is probably the closest thing I could find:

There is disagreement even on his name. Garth Greenhand, we call him, but in the oldest tales he is named Garth Greenhair, or simply Garth the Green. Some stories say he had green hands, green hair, or green skin overall. (A few even give him antlers, like a stag.) Others tell us that he dressed in green from head to foot, and certainly this is how he is most commonly depicted in paintings, tapestries, and sculptures. More likely, his sobriquet derived from his gifts as a gardener and a tiller of the soil—the one trait on which all the tales agree. "Garth made the corn ripen, the trees fruit, and the flowers bloom," the singers tell us. -TWOIAF: The Reach: Garth Greenhand

and:

The green men ride on elks, Old Nan used to say. Sometimes they have antlers too." -ASOS, Bran IV

Think so?

Lommy Greenhands was obviously a member:

Arya made the stick whistle as she laid the wood across his donkey's hindquarters. The animal hawed and bucked, dumping Hot Pie on the ground. She vaulted off her own donkey and poked him in the gut as he tried to get up and he sat back down with a grunt. Then she whacked him across the face and his nose made a crack like a branch breaking. Blood dribbled from his nostrils. When Hot Pie began to wail, Arya whirled toward Lommy Greenhands, who was sitting on his donkey openmouthed. "You want some sword too?" she yelled, but he didn't. He raised dyed green hands in front of his face and squealed at her to get away. -ACOK, Arya I

Terrible joke. I'm sorry lol

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TLDR: Everything (not much) we know about the Order of the Green Hand.