There is a difference between improving the fortunes of an impoverished community, and forcing that community elsewhere.
I went to the Five Points once, and it was unrecognizable. Though New York changes quite a bit, the city now changes less than the old city used to. The existing buildings are massive high-rises (I think housing?) that are clearly quite wealthy, and a municipal building that serves the borough of Manhattan. One approach to consider would be an alternate history backstory even if the story takes place in the future, because I think the window to recreate the Five Points has passed.
The clear trend to butterfly away would be the Urban Renewal ascendence of the mid-twentieth century. It is wrapped up a lot in modernism as an ideology and an architectural school, and I suspect the privileging of engineers and technocrats in the Second World War also contributed. The original plans for interstate highway systems were to include rail corridors, and split into boulevards when they entered the city center; it was not until engineers--Robert Moses being the ur-example--took control of implementation and focused almost exclusively on traffic flow that the concept of a highway serving the city was abandoned. Instead, the city was something that needed to be wiped clean to let the highway pass through (or the community center, or the housing project, or the office block). Thus, mass evictions and slum clearances wiped clean much of the slate that cities had been built on. Lower Manhattan would have been cut through by a ten-lane, four-interchange expressway through Greenwich Village, Soho, Little Italy, and Washington Square were it not for the community organizing efforts of Jane Jacobs and her neighbors.
This was part of a broader trend of hollowing out American cities and pouring investment and incentives into suburbs and sprawl. This is why American cities became so dysfunctional in the 1970s and 1980s. Gentrification is essentially a return of those suburbanites, or their children, back into the cities now that city life is in demand once more. The hostility towards it is provoked by the fact that the people who have actually been living in those neighborhoods this whole time are being passed over and forced out after generations of neglect and hostility. "Who told you to buy a brownstone in my neighborhood
on my side of the street?!"
How to butterfly away this trend of urban renewal and suburbanization is a huge question that I don't have an answer for, though it's a favorite topic of mine. What we are looking for, I think, is a continuation of the tradition of Daniel Burnham's City Beautiful, and the old tradition of civic design. We would need to butterfly away the death of the streetcar and passenger rail (and save Penn Station!) to create fewer incentives for suburbanization, and we would need to stave off the temptation to undertake massive megaprojects in favor of what Jane Jacobs called "close-grained diversity," or small lots of mixed incomes and uses.
It's a tall order, but that's the kind of world that might have restored the Five Points sometime in the twentieth century. Perhaps the name will come back someday--goodness knows we're fascinated by the place--but I suspect that the built form that the Five Points was will never return. Creating a Five Points that could exist, because it was rebuilt sometime in a different past, would not only mean the creation of that neighborhood, but of an entirely new city. In this city, the New York of memory, or parts of it, would still exist, alongside the wonders of a future we'll never see.