They are not alive, but do orbit unpredictably.
Dragon magazine #174's Bazaar of the Bizarre article on ioun stones describes that ioun stones orbit seemingly randomly, following an individual's magnetic force lines. They aren't intelligent as such, but sometimes move as if they are.
By firmly grasping a stone, any character can imprint his personal magnetic aura into the gem's structure. The activated crystal is then tossed lightly into the air, where it orbits the user like a small glowing moon. This orbit is not always elliptical but follows a seemingly random, meandering pathway as it maneuvers along the user's magnetic force lines. A stone alternately sits motionless in midair or dances and weaves about, rarely straying more than 3' from its owner. Sometimes, a stone wanders up to 12' from its user for no apparent reason, but it quickly returns, its obscure "Mission" completes.
It also mentions that they can appear to move based on the bearer's thoughts, to a limited extent:
Some wizards have observed that if a character is looking intently at something, some ioun stones that he possesses wil move toward the object of his scrutiny.
This article was written during the AD&D 2e era of D&D, though its lore can still be applied to D&D 5th edition. There is relatively little 5e-specific lore on ioun stones due to the fewer books published and lack of magazines to cover minute details such as the exact orbit of an ioun stone.