The colloquial "Briar Patch" name used in the Federation was first coined by criminal geneticist Arik Soong in the 22nd century, a reference to the Br'er Rabbit of the 1880s Uncle Remus stories written by Joel Chandler Harris of Earth.
This region was known for containing dangerous space matter, including the remains of supernovae, false vacuum fluctuations, metaphasic radiation, and at least two habitable planets, one of which was settled by the Ba'ku. 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 preferably were specially fitted for it. (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 2271, the region was the site of the celebrated Battle of Klach D'kel Brakt, where 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)
Background information Edit
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)
When Star Trek: Star Charts was published in 2002, the relationship between the Briar Patch and Klach D'kel Brakt had not yet been established in canon. ("The Augments", which established the connection, wasn't aired until 2004.) Star Charts (p. 63) depicted Klach D'kel Brakt as a star system in the Beta Quadrant. The more recent "Stellar Cartography" ("Stellar Cartography" p. 17) briefly mentioned that the Briar Patch and Klach D'kel Brakt were names, Federation and Klingon, respectively, for the same supernova remnant. There was no depiction of the Klach D'kel Brakt system on any of the charts.
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, suggesting that the two aren't always synonymous.
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.