I would change the name to poor lives matter.
Black Americans have it much better than they did in 1860 and 1960, yet they still don't have the same opportunity as white Americans in 2016. They are still being killed by police in questionable circumstances disproportionally.
The issues still need work. Black Lives Matter, is working for more justice in that regard. I have a problem with the name in that it gives the Alt Right an argument that they are being reverse-racists. I don't have a problem with what I know about their work.