Einstein became an international celebrity after the story of the eclipse spread. He remained in the public eye for the rest of his life.
But Eddington was also well known during his lifetime. Stellar physicists would still know his name today because of the "Eddington limit," which is the maximum luminosity a star can have when it's in equilibrium. But in his day, besides his academic work, he was a science popularizer. He wrote popular science books and gave lectures to communicate scientific ideas to the public.
Eddington also achieved the ultimate 20th century accolade - in 1934
Eddington made the cover of Time magazine!