The colloquial "Briar Patch" name used in the Federation was first coined by criminal geneticist Arik Soong in the 22nd century, a reference to an Earth story involving Br'er Rabbit. (ENT: "The Augments")
This region was essentially a gas cloud, known for containing dangerous space matter, including false vacuum fluctuations, metaphasic radiation, and at least two habitable planets, one of which was settled by the Ba'ku. As well it was flooded with radiation from supernova remnants.
Ships in the vicinity had to travel at less than one-third impulse power to avoid overheating the vessel's impulse manifolds. According to Admiral Dougherty, ships entering the region usually were specially fitted for that purpose. (Star Trek: Insurrection)
In May of 2154, Soong suggested the cloud as a hiding place for himself and a group of Augments aboard a stolen Klingon Bird-of-Prey, but his suggestion was rejected by Malik, the self-appointed leader of the Augments. (ENT: "The Augments")
In 2257, Klingons occupied the star Ba'ku's system. (DIS: "The War Without, The War Within") The same year, the location was labeled on the star chart showing the trajectory of Lieutenant Spock's shuttle. (DIS: "An Obol for Charon")
In 2271, the region was the site of the celebrated Battle of Klach D'kel Brakt, during which Dahar Master Kor led the Klingon Empire in a glorious victory over the Romulan Star Empire. Nearly a century later, Kor was still fond of re-playing the battle in holosuites. (DS9: "Blood Oath")
The anomalies pervading the region had not been fully identified when in 2375 a joint Starfleet-Son'a mission was mounted to study the Ba'ku planet. During that mission, the Briar Patch was the scene of the Battle of the Briar Patch between the Federation starship USS Enterprise-E and Son'a ships, following the discovery of a plot to relocate the Ba'ku. (Star Trek: Insurrection)
The names of the Briar Patch's stellar contents and neighbors originate from a computer-displayed okudagram which Geoffrey Mandel created for Star Trek: Insurrection. While not clearly seen on-screen, the graphic was reproduced in The Secrets of Star Trek: Insurrection (p. 143) by Terry J. Erdmann.
According to the Star Trek Encyclopedia (4th ed., vol. 1, p. 103), the Briar Patch was "whimsically named for the tangled hardwood shrub that featured prominently in the Uncle Remus stories by Joel Chandler Harris, published in 1880 and 1883."
In the script of Star Trek: Insurrection, the Briar Patch was described as a "dangerous and beautiful region." 
For Insurrection, the appearance of the Briar Patch was developed by Mark Wendell, CGI supervisor at Santa Barbara Studios, and Art Director Richard Kriegler. "Our goal was to make this area of space look like a big nasty rat's nest, filled with gases and clouds and so forth," Wendell explained. (The Secrets of Star Trek: Insurrection, p. 26)
According to the Star Trek: Star Charts (p. 63), Klach D'kel Brakt was a multi-star with a G-class star, an M-class star, a B-class star, and an F-class star.
The star chart used as a source in Star Trek: Discovery was taken from the Star Trek: Star Charts, which was first published in 2002; the connection between Klach D'kel Brakt and the Briar Patch was made in the Star Trek: Enterprise episode "The Augments", which was first aired in 2004. As such, this caused some disparity, as Klach D'kel Brakt was shown as the name of a location outside of the Briar Patch in the map's appearance in Discovery, suggesting that the two aren't always synonymous.
The more recent Stellar Cartography: The Starfleet Reference Library (p. 17) briefly mentioned that the Briar Patch and Klach D'kel Brakt were names, Federation and Klingon, respectively, for the same supernova remnant. The chart in the episode "Magic to Make the Sanest Man Go Mad" labels the Klingon name "Klach D'kel Brakt" as a location distinct from the Briar Patch,
According to the RPG sourcebook The Way of D'era, Klach D'kel Brakt was a mineral-rich planet known to the Romulans as Dumok'azen.
In the miniseries Star Trek: Prey, Worf is briefly captured by the descendants of the discommendated followers of Kruge, known as the Unsung, as they see him as a powerful symbol due to his own history of suffering discommendation and then regaining his honor, taking him to a planet in the Briar Patch. Worf is able to use his own experience of the location to alert his allies to the area to search during a later broadcast by the Unsung, stealing the armor of one of their number and fixing the ace of clubs from a stolen deck of cards to his weapon as he appears in a broadcast to the rest of the quadrant, recalling a unique stellar formation in the Briar Patch that Riker compared to that card and hoping that one of his crew will see it and realize the clue.
The Briar Patch appears in three missions in Star Trek Online. In "Secret Orders", Federation player characters are sent into the Briar Patch to locate and destroy a Klingon weapons research facility operated by the mad Ambassador B'Vat. In "Brushfire", following the Iconian War, player characters of all factions are dispatched to rescue former Chancellor Martok from a prison operated by the Son'a and the discommendated House of Torg, before Torg can turn him over to the Tzenkethi. Finally, in the patrol mission "Within the Briars", part of the ongoing Age of Discovery storyline, the holographic Paul Stamets directs the player character to shut down a mycelium spore harvesting operation in the Briar Patch run by the temporally-displaced House of Mo'Kai.