What were the Browns and other families asking the Supreme Court to do

The case in question is "Brown vs. Board of Education" - and it dealt with the question whether the racial segregation of schools is lawful - with the decision reached being "no" - this case ended the state-organised racial segregation of schools.

Among the options, the best answer is "Allow African American children to attend the school of their choice" - so that they could attend the school which up to then were "white" schools.