(covers information from several alternate timelines)
In 2293, the location of Rigel in the Milky Way Galaxy was labeled in a star chart that was in Captain James T. Kirk's quarters aboard the USS Enterprise-A. (Star Trek VI: The Undiscovered Country, okudagram)
In 2366, in an alternate timeline, during the Federation-Klingon War, the location of Rigel was labeled on a tactical situation monitor in the ready room aboard the USS Enterprise-D (TNG: "Yesterday's Enterprise", okudagram)
|Rigel II • Rigel III • Rigel IV • Rigel V • Rigel VII • Rigel X • Rigel XII • Asteroid Belt|
In the real world, the star Rigel was a star located 773 light years from the Earth in the constellation Orion While this constellation was seen on screen in TNG: "Family" and on a constellation star chart in the TOS: "The Corbomite Maneuver". The Rigel in Star Trek was apparently not a star named by Humans, since Vulcans and Klingons used this name in their native languages and the location of Rigel was not known to Humans before it was learned from the Vulcan star charts in "Broken Bow".
According to The Worlds of the Federation (pg. 36), the Rigel system and the Orion system were one and the same. According to the reference book Star Trek: Star Charts (pg. 36), there were in fact two stars called Rigel. These were Beta Rigel and Beta Orionis. The Rigel system was in Beta Rigel. According to Star Trek Maps (Chart D), the Rigel system was in Beta Orionis.