I am wondering if there is some example of a mathematician or physicist who published other papers at the same time as their PhD work and independently of it which actually eclipsed the content of the PhD thesis.


The only semi-example I can think of immediately is Einstein, whose other publications in 1905 (especially on special relativity and the photoelectric effect) eclipsed his PhD thesis which was published in the same year. Although it contained important insights, it was somewhat forgotten to the point where he felt that he had to remind people about it.


Although this is a soft question, I didn't ask in Academia as I didn't want examples outside of mathematics and physics.



Example of a Mathematician/Physicist whose Other Publications during their PhD eclipsed their PhD Thesis
[Anatoly Karatsuba](https://en.wikipedia.org/wiki/Anatoly_Karatsuba) discovered the 
[Karatsuba algorithm](https://en.wikipedia.org/wiki/Karatsuba_algorithm) in 1960, and reported it to Kolmogorov who published it under his (Karatsuba's) name without his knowledge. It seems fair to say that this first example of a "divide and conquer" algorithm eclipsed Karatsuba's 1966 thesis on "The method of trigonometric sums and intermediate value theorems".


For a physics example (from my own university) I note [George Uhlenbeck](https://en.wikipedia.org/wiki/George_Uhlenbeck), who with Goudsmit introduced the electron spin in a 1925 publication, while his 1927 Ph.D. thesis on quantum statistics was much less influential. ([Here is the story](https://ilorentz.org/history/spin/spin.html) how two Ph.D. students discovered the electron spin, which was missed by a giant like Pauli.)


