Planets' rights were rights held by individual worlds.
According to Beckett Mariner, judge Mith bin Tong was a "planets' rights lunatic" (i.e. someone who strongly believed in planets' rights), which was bad news for Mariner's mother Carol Freeman, who was about to be tried by the judge. (LD: "Grounded")
This concept would appear to correspond with the real-world idea of states' rights.