The case for gentrification is usually something in terms of economic development. The notion that we shouldn't halt development, halt progress, just because some people lose out. Other people gain from that same progress, and it advances both the neighborhood and society as a whole.

Some people also make property rights arguments in favor of gentrification, I mentioned this is another comment. Basically: our system rewards those who have the most money, and gentrification is simply one example of that. And this is good, because our system is good. Or at least better than all other options. (people who make this kind of argument never actually examine all other options)