Problem that anyone who tries to describe an image from centuries ahead always faces: at what point will your present still exist there in the future?
I can think about how fashion and art are cyclical and how a reduction in population will also cause a reduction in artistic innovation, in addition to reductions in other areas.
But, this does not solve the problem. It is not enough. Why will Charles Mingus, Tom Jobim, Leonard Cohen or Fergie - among others - still be heard in 200 years, not by an erudite audience (like classical music lovers), but by an ordinary audience? What kind of movement could bring these cultural elements all back into everyday life 200 years later?
In history there is a period of near-extinction where 2/3 of Humanity disappears. However, for the survivors and descendants of these later the world becomes much more pleasant. The memory of the Great Depression between 2030 and 2050 is still a trauma that did not disappear by 2220~2240, however the situation is more of a post-scarcity economy, even more so with the colonization of Mars, space mining and more efficient use of energy and resources.
There are still no means for interstellar travel.
Of the 4 billion humans, about 900 million are on Mars and another 600 million are on space stations, most in the orbital rings of Earth and Mars, and a few on errant stations from the regions of Venus to the orbit of Jupiter.
All the answers are great and taking on interesting points, but I think I need to explain some additional things:
The depression period does not come with a major world war and ICBMs nuking important cities on the other side of the planet. National states degrade, collapse and implode. Of course, eventually a nuclear bomb will fall into the hands of some bad guys or desperate and immoral governments will use it against insurgents from the own population, but nothing that will create a widespread blackout so fast. The globalized Internet will fragment into dozens (hundreds? thousands?) of smaller networks. Harm caused by chemical and biological attacks, the consequences of climate collapse, apocalyptic doomsday cults and the emergence of warlords controlling diverse regions are all part of the scenario. Also, there are tons of media files on hds from nostalgic p2p nerds and piracy. It won't be necessary to dig a bunker at the end of the world to recover the Stones' catalog or Beethoven's symphonies.
The devastation caused makes generations growing up in depression more, um... politically responsible. They organize their villages and later their city-states as closely as they can to direct democracy. This brings 200 years of peace and prosperity to the humankind, to the point of building space elevators and terraforming Mars. Certainly the governmental elite developed its methods to keep society tamed, but nothing like the methods of coercion.
This is the problem that made me come here to ask the question: people work less and less, have more pleasant lives than humanity has had so far, with plenty of energy, water, resources that are well used and reused, and space. In these 200 years, there are two great moments of creativity explosion and the story takes place on the verge of a third.
The names I mentioned are specific preferences of some characters who have different ages and origins. It is just as unlikely that 3 individuals who are born in very different places on the planet have a preference for names from the music of 220~280 years earlier than we have today people who like and have strong opinions about Mozart and Bach, without some cultural force provoking it. Of course, other obvious names that were mentioned in the answers and comments will also be remembered, among others. But why?
After disasters where a significant portion of the population dies, the quality of life of survivors increases. Climate collapse will inundate cities, desert farmland and diminish the availability of fresh water, but all the existing infrastructure that once supported 8.4 billion humans by 2029 will be harnessed by 2.9 billion by 2050. These 2.9 billion also know that the inefficient and resource-wasting mode doesn't work and... nuclear fusion! Since all evolution is measured by energy availability, people in post-depression will be able to achieve a lot very quickly, on the one hand. On the other hand, only 1/3 of the population means fewer brains dedicated to innovation.