There are two main types of benefits that we might arguably (we cannot, of course, be sure of the consequences of such a change) gain from doing affirmative action on the basis of income rather than on the basis of race.
First, we might reduce the inequality between rich and poor. In recent times, it has become much harder for people to move up the socioeconomic ladder in the United States. This makes it much harder for us to see ourselves as the land of opportunity since it is much more common for people to be stuck in the lower classes if they are born there. If we did affirmative action on the basis of income, we might make it more possible for people born in the lower classes to improve their lots.
Second, it is possible that we would improve race relations in the United States. With our current system of affirmative action, we may arguably be making race relations worse. We are creating a situation in which whites feel that they are being treated unfairly and, at the same time, minorities feel angry that whites want to do away with affirmative action. If we based affirmative action on income, we would still help poorer minorities without causing the kind of racial ill-will that is generated by the current system.
Thus, we can argue that income-based affirmative action would improve our levels of social mobility as well as our race relations.