Elon Musk’s SpaceX, demonstrating its founder and chief executive’s penchant for showmanship, announced that it had signed up the first private passenger seeking to fly around the moon. But the company provided no timetable or other details about the plan.

In a message on Twitter on Thursday, closely held Space Exploration Technologies Corp. said the mission is slated to use its largest rocket, dubbed the BFR, which is still under development and, according to some industry officials, may be at least a year or two from an...