The name of Rigel is used because Rigel is one of the best known star names and one of the stars science fiction writers are most likely to have had pointed out and named to them in the night sky. So early generations of science fiction writers used the names of Rigel and other well known stars and later generations of science fiction writers copied that practice because it was already a trope.
It should be noted that the practice of setting space opera stories on planets of stars well known to the readers is rather illogical. The stars that are bright enough as seen from Earth to be visible to the naked eye number less than 10,000. those are the stars that have proper names like Vega or Deneb or Antares; or Bayer constellation designations like Beta Geminorum or Sigma Draconis or Omega Cygni or Alpha Carinae; or Flamsteed constellation numbers like 40 Eridania or 61 Cygni.
They range from rather ordinary main sequence stars that happen to be close to the solar system, like Alpha Centauri the closest star system only 4.3 light years from Earth, to giant stars like Aldebaran and Arcturus tens of light years from Earth, to giant stars, bright giant stars, and super giant stars hundreds of light years from Earth, to bright giant stars, super giant stars, and hyper giant stars thousands of light years from Earth.
According to this question, the most distant naked eye star is probably less than 10,000 light years from Earth.
In our region of the galaxy there are about 0.004 stars per cubic light year.
So a sphere with a radius of 10 light years centered on Earth would contain 4,188.79 cubic light years and thus about 16.755 stars.
So a sphere with a radius of 100 light years centered on Earth would contain 4,188,790 cubic light years and thus about 16,755.16 stars. Even if the less than 10,000 stars visible to the naked eye from Earth were all less than 100 light years from Earth they would be less than 0.596 of all the stars within that distance.
So a sphere with a radius of 1,000 light years centered on Earth would contain 4,188,790,000 cubic light years and thus about 16,755,160 stars. Even if the less than 10,000 stars visible to the naked eye from Earth were all less than 1,000 light years from Earth they would be less than 0.000596 of all the stars within that distance.
Assuming that the galactic disc is about 1,000 light years thick near Earth, a squat cylinder with a height of 1,000 light years and a radius of 10,000 light years should have a volume of about 3.14 times 10 to the 11th power cubic light years and thus about 1.256 times ten to the 9th power stars, or about 1,256,000,000 stars. If all the less than 10,000 stars visible to the naked eye from Earth are less than 10,000 light years from Earth - and they probably all are - they would be less than 0.0000079617 of all the stars within that distance.
Since the galactic disc has a radius of about 50,000 to 60,000 light years, the part of the galactic disc that is within 10,000 light years of Earth is only about 0.04 to 0.0277 of the total volume of the galactic disc.
So in a story set in a galaxy-wide civilization any person's home planet would have only a 0.277 to 0.04 probability of being within the section of the galaxy where stars visible from Earth are found. And any star that a was within that small region of the galaxy would have less than a 0.0000079617 probability of being a star visible to the naked eye from Earth. And of the less than 10,000 stars visible to the naked eye from Earth, probably less than a hundred hav names familiar to most people on Earth.
So stories where all the important stars have names familiar to Earth readers would be fine if set in a fictional imaginary galaxy that only has one hundred stars in it. But they are not so realistic when set in the real galaxy that has a billion (1,000,000,000) times that many stars!
Another thing that I noticed when a kid reaching science fiction and studying astronomy is that more and more of the named stars have measured distances from Earth, distances that keep getting measured more and more accurately. In stories about a star ship exploring the stars, it seems desirable for most of the stars it explores to be at the frontier of exploration, and thus about the same distance from Earth.
And many science fiction writers have no idea of the distance of particular named stars from Earth, and mention names that lead to galactographical inconsistencies.
Furthermore, many famous bright stars are of spectral types that are not likely to have habitable planets or native advanced lifeforms. Thus it might be more plausible to say something like "planet three of Rigel sector star 182l" or "planet Stanortul orbiting a star in the Antares sector" or "Amutere near Deneb" instead of "Rigel III", Antares IX", or "Deneb VII".
There is another group of stars whose names or designations are well known. They are stars that are very close to Earth. Many of them are so dim that they are not visible from Earth even though among the closest stars, and thus don't have names but star catalog numbers, often several different ones for the same star.
in order of distance from Earth they are Alpha (and Proxima) Centauri, Barnard's Star, Luhman 16, WISE 0855-0714, Wolf 359, Lalande 21185, Sirius, Luyten 276-8, Ross 154, Ross 248, Epsilon Eridani, and somewhat farther away: Procyon, 61 Cygni, Epsilon Indi, Tau Ceti, Kruger 60, etc.
Many science fiction writers have read such lists of nearby stars and so they appear in some science fiction stories.
for example, L. Sprague de Camp's Viagens Ineterplanetarias stories starting in the late 1940s are set on planets of nearby stars such as Lalande 21185, Procyon, Epsilon Eridani and Tau Ceti.
And their use was copied by other science fiction writers who have no idea how close or far away they are.
And this can lead to very inconistent mixes of star names (and thus of distances) in some stories.