I'll copy-paste directly from my own idea notes (these are set in the somewhat EU universe where Luke has established the Academy):
Enemy cult, Children of the Void: a spacer-related cult of Force-users, relying heavily on scavenging. While originally a cult intent on studying the universe itself trough the Force, they have been attracted to the Dark Side trough the void's emptiness and entities within. They have a history of being connected to the old Order's Exploration Corps but not exclusively.
They believe that they are in contact with powerful entities that they reach trough the Force, entities that they believe is not simply a more alien form of life but the Force manifested in sentient clusters. These entities are real (although not what the Cult think of them as) but are difficult to contact and they deliberately avoid contacting Jedi or Sith, both sometimes sensing their presence but unable to contact them. The link between these entities (I haven't named them yet) is strongly telepathic and has strong influence over their followers, changing them and teaching them.
A further link is an alien species commonly known as Deep-dweller, a species adapted to vacuum life and with a super-low temperature biology, who are the oldest members of the cult and have Force-sensitive members. They are extremely alien compared to most Republic species and their experience the Force differently, although they still have Light and Dark side. They are capable of temporarily joining their brains together to form temporary joined minds.
What the entities actually are I am not sure yet. They would have physical aspect to them and would be truly ancient.
The cult's religion is complex, heavily centering around the entities and trying to ascend their biological body. They view the existence as life as merely an accidental by-product of specific environments and view various cosmic processes as what is actually important to the world. They live mostly on spaceships that are well-kept and all modified to be self-supporting, often with zero gravity and feel that going on a planet's surface is some sort of trap. They often have extended-stay space suits that they train to be able to stay in for months. Cybernetic implants and augmentation is common, although a significant portion of them is devoted to allowing them to live as spacers. The cult also focuses on meditating on the void around them, commonly trough hyperspace. The cult is very esoteric and closed, only accepting members they send invitations to. Unlike the Jedi and Sith, the cult encourages the creation of children and views the cult itself as a large family unit. Children are not exclusively raised by their parents but by the entire cult, every member encouraged to treat any child as their own. This makes all ties to the cult strong family ties.
One important aspect of their power is the knowledge of ancient Force-enhanced technology, Rakatan or otherwise. The cult is apt at finding the usable remains of ancient civilization. They hoard such technology and are dogmatic about not sharing it, viewing it as holy.
The cult prefers economic power over military might, but their fleet of starships allow them to project violence. However the cult is not devoted to the Dark Side and uses it in an unusual fashion. Rather than ego, fear and anger, they use things like mania, religious fever aided by drugs and cybernetics to have controlled bursts of insanity that makes their relationship with the Force different. When not using such the Cult also is able to use the Force to see trough the void, able to find and detect things that fit their interest, especially wrecks and starships lost trough hyperspace accidents. They never simply abandon a wreck, always able to scavenge and salvage. They are also able, trough specialized hardware and hyperdrives, to influence hyperspace travel and thus able to use hyperspace routes that would be otherwise too dangerous to use by regular crafts.
The cult is broken down to many clusters and cells that are largely independent from each other. Hierarchy is broken down within a cluster by seniority in being contact with their entities and their relationship with each other. These leaders pledge themselves to the service of the cult's worshiped entities and trough them, have contact with each other. Some clusters are more friendly to outsiders than the others (typically closely tied to how much they trade), their leadership varying cluster-by-cluster. A cluster is always tied to a fleet or some fixed location, each dividing up hierarchy among themselves. An individual's rank is closely tied to what spaceship they posses and what role it has within a cluster (the larger the better).
They are apt at zero-g combat and like to force their opponents to using it to gain an upper hand. They prefer to scavenge various weapons as they use the Force to find any usable and suitable weapons as "gifts of the void". Lightsabers are preferred, especially ancient ones that make a black core, but any variety of weapons is used. The cult does not view physical alternation or Force-aided combat abilities as high-priority. All of them are capable pilots (being able to pilot considered a sign of adulthood) and all combat-oriented members have their personal, customized starfighters.
The Cult in the past is is secretive and stays away from Galactic governments and their influenced areas, be it the Republic or the Empire. This greatly aided in their survival. The Old Jedi Order knew of their existence but back then were considered just a minor cult and the Exploration Corps kept them in check. During the Empire they were hunted but were considered a low-priority threat, lower than that of the Rebellion. The Cult's mobile and clustered nature also helped them, simply moving itself away from Imperial space and thus away from their influence. Palpatine wanted some of the cult's secrets and considered them a low-key rival. He knew of the entities that the cult centers around (via a previous Bane-Sith that investigated them) and at one point managed to contact them, but they required submission of a person to gain their power, something that Palpatine obviously refused.
The Cult has its own network of infrastructure, all mobile when possible and well-defended and/or well-hidden when not. They have a significant fleet of found, restored and self-built spaceships. For labour they rely on droids and automation, and on non Force-sensitive members. What they can't produce themselves, they will trade, sometimes selling mapped hyperspace routes or scavenged supplies. Several more powerful cult members are traders, uncaring of laws but unwilling to transport anything alive (be it pets, livestock or slaves) due to how they view their spaceships as their personal space.
With the falling apart of the Galactic Empire, they have become a power in the vacuum that the Old Jedi Order and Palpatine's own Sith cult have left (there were other Force-sensitives under Vader and Palpatine such as the Inquisitor), they have allied themselves with several Imperial warlords or systems unwilling to join the Republic. Another aspect is that with the death of the Trade Federation, they are rising as a legitimate trading and Force-using political power trough the Galaxy, even within the New Republic. While the New Jedi Order is building itself, the cult has survived mostly whole and outnumber the Jedi.
We see how the Jedi dominate and hide, how they fight a hostile opposite of themselves, but we haven't seen them much of them handling legitimate rivals.
But the problem with adding a new threat to Star Wars is that the galaxy is one that is already explored and well understood.
There are still large regions of the Galaxy that are unexplored, according to the old guides and atlas.
But the problem with adding a new threat to Star Wars is that the galaxy is one that is already explored and well understood.
Unless it is outside of their reach or was not a threat during the Clone Wars and beyond.
Another unrelated idea I mentioned in the other thread was that of Jedi vs Jedi. In the early days of rebuilding, what if Jedi were split on the issue of justice vs peace. One camp argues that the function of the Jedi should be striving for justice whatever the cost, pointing out that the Jedi of the Old Republic lost because they were too strongly tied to the poltics of the Old Republic. The second argues that their more important function is maintaining peace and working with the New Republic, pointing out that Jedi are not capable of operating on their own in their current state.
The justice cause does not make sense to me as you present it: how do they propose to instil justice of any kind without the New Republic? For there to be justice there needs to be some sort of societal value system to which according to justice works by. Without the Republic supporting them and giving them authority, a Jedi is just a citizen with a lightsaber (if that) and the Jedi Order a cult. Are they supposed to just go to whatever world and throw themselves at the first case of injustice they find? Essentially becoming renegades that only obey whatever law they decide to hold themselves to, ignoring the larger political picture and dictating to society as they see fit?
In fact without the Republic supporting them there is no Jedi Order. The New Jedi Order needs funds to operate, to buy their spaceships, to buy the food they eat, to get, maintain and supply the spaceships they use, etc. Even the old EU stuff about the Old Jedi Order concede that they must have credits if nothing else. How does the New Jedi Order fund itself if not trough the Republic, especially as it grows in numbers? If they make any sort of basis for income they will be tied to that. If they tie themselves to donors they will have to prioritize those donors. If Jedi can be hired than they essentially become mercenaries of a sort.
You are right however that the New Order's redefining themselves would have the question of whether it should devote itself to being agents of justice or peace. I would also add how they would have to redefine their relationship to the Republic, whether to integrate themselves to existing branches of the Republic or put distance from them. There is also the question of authority, should the Jedi be independent agents answerable only to themselves and the Senate, empowered to do whatever they think is in needed? Or should they be more humble and merely assist existing branches of the New Republic?