The idea of turning Jupiter into a second sun is an old one, going far back to the 20th Century. The idea became popular in the science fiction novel 2010: Odyssey Two. The idea gained traction near the turn of the tenth millennium when a team orbiting a J-Class planet Occam managed to successfully 'ignite' the planet and turned it into a dim brown dwarf, which in turn made a few of Occam's moons habitable and terraformable.
With the success of Occam's Star, many others began refining the technology and successfully ignited other suitable J-Class worlds, including a few rogue planets. On July 5, 9999, a motion was put on the ballot during the Federation campaigns that would allow for the ignition of Jupiter. The ignition would mean that heat would warm the four Galilean moons more naturally.
Controversy surrounded the proposed project. Many who opposed the project pointed to the Metallica Incident in which a botched ignition caused the disruption of the entire star system, including a populated human colony world. Many of the planets in the system were thrown out of orbit. Others pointed out that Jupiter was a historic landmark, so to speak. It was one of the oldest worlds known to humanity and many believed igniting it would be akin to destroying something like the old Statue of Liberty on Earth.
Due to the controversy and opposition, the project was scrapped. Instead, the technology was used to continue and ignite many large gas giants, creating new inhabitable worlds out of the moons.