It's hard to identify why someone's fame isn't lasting. It's the most common occurrence — fame that lasts well beyond your own death is unusual, even in science where, more than many other fields, the highest mark of esteem is getting something named after you and being integrated into the "canon" of "famous scientists." Everybody knows Galileo; almost nobody knows Christopher Clavius, who was arguably the most famous astronomer at the time Galileo was alive (he was the top Jesuit astronomer, the guy who wrote the textbooks that you'd be using if you were studying astronomy in the mid-17th century). Galileo became the symbol of the new science, of science vs. religion, etc. etc., whereas Clavius became, at best, a subject of historical interest to people who are really wanting to dive deep into the world of 17th-century astronomy.

Sometimes scientists get and stay famous for rather unseemly, or at least "non-scientific," reasons. John Bardeen won two Noble Prizes in Physics (the only such person to do so), but is basically unknown, even to physicists, because while he was smart and accomplished, he was un-eccentric. His co-Nobelist, William Shockley, is much better known... but not for good reasons (Shockley became famous as an almost monumental jerk, both in his treatment of his employees, and for his views on eugenics). Would non-physicists still talk about Richard Feynman if he didn't play the bongos and write humorous books? Maybe — but if you strip away the eccentricities, the odds of him being a "public figure" after his death get a lot lower, even with his other public contributions (e.g., his work on the Challenger panel — can you name anyone else on that panel? If you look up the names, you'll recognize a a number of them, but I'll bet you didn't know they were on the panel before you looked it up.).

One might also suggest that Millikan had the unfortunate luck of being a major physicist in the 1930s, but "out of the game" by 1945 (he moved from research into university administration by the 1920s; retired in 1945; died in 1953). This happens to coincidence with some major shifts in public knowledge about physicists, brought about by World War II and the atomic bomb. In other words, at the same time his star was fading, the stars of many others burned very brightly: a "Golden Age" of physics was beginning, with a huge amount of attention, funding, and prestige being showered on the generation after Millikan, leading to huge advances in the field (quantum electrodynamics, solid state physics, transuranics, etc., to say nothing of practical advances with fission, fusion, semiconductors, etc.). Millikan's own scientific contributions, while important for their time, seem very old-fashioned in comparison (e.g., measuring the charge of the electron feels very "early-20th century" rather than "mid-20th century"; it is a pre-WWII form of physics and physical research).

All of which is to say: there is nothing particularly surprising in Millikan's loss of public awareness (there is no need to concoct an elaborate explanation, in other words), but if one were looking for elaborate explanations, the "timing" of his years of fame is such that one might not be surprised to find he was quickly overshadowed.

On the "Golden Age" of physics and the role of World War II in particular in changing how physics was perceived, see esp. Dan Kevles' The Physicists.