For example, in the Soviet Union the names of the average person were still most likely the name of some saint or apostle. Konstantin, Ivan, Andrei Nikolai etc etc etc. Even in modern Cuba you still see Juan, Mario, Nicolas, Jesús...doesn't this go against state atheism policy?
Shouldn't we see names like transliterations of communist figures or at least secular names, especially after decades upon decades of communism rising and falling in the world?
As anti-religious as the Cuban Revolution was in its first few decades, I know of not a single radical Cuban Communist to even propose this.
Names like José, María, Juan, etc., were just normal names for people of Spanish descent (or at least those who were culturally hispanic). While the very religious would name their kids according to whose Saint Day it was, many would name kids to honor family traditions (such as men in the family always being called Benito) or to honor friends of the family. These people would also tend to be called traditional names.
In Cuba you do have ideologically motivated names, but such things are rare and when these kids grew up they often hid their names and used nicknames to transform the hideous names into something more acceptable.
The big secular name boom of the 1980s, known as Generation Y, was typified by people with strange names unrelated to traditional and religious ones. Dissident blogger Yoani Sanchez is one example. There was also an infinite number of variations of Misladies, miladies, which were poor attempts to copy a variant of 'my lady' fedora tipping intensifies. Other names, such as variations of Île du Lys (Laidelis, Lideliz, etc.), caught on because of documentaries and the like. Still other names were inspired by characters from telenovelas, in particular the ever popular Brazilian telenovela.
To my knowledge, however, no conscious effort to eradicate traditional names ever took place. Likely because Cuban politicians recognized that these represented merely nominal nods to Catholicism and that they had personal significance which would be tough to stamp out; how, for example, do you tell someone they can't name their kid José despite all the men in their family, going back generations, having José somewhere in their name? In the end, such a project is also patently ridiculous, which also helps explain why it was never implemented or attempted.
The things that were attacked were a) religious holidays, which ceased to be recognized (only becoming official holidays again through the visits of John Paul II, Benedict XVI, and Francis I), b) religious people joining the Communist Party (which only became acceptable again in the 1990s), and c) the Catholic Church in that it is the best financed and most organized group of any sort operating outside of Cuban government control. The worst period started in 1971, with the Congress on Education and Culture, and saw things start to get better in 1991.