While there were anti-segregation activists in the 1950s and 1960s, none of them were leaders of the movement so they are not famous. The first link I've attached is a picture from the March on Washington showing various white people in the crowd around one of the speakers. You could call them anti-racism activists since they are part of the protests, but they're not famous.)
Perhaps the most famous white activists were Michael Schwerner and Andrew Goodman, who were killed in Mississippi during an effort in 1964 to register blacks to vote.
Historically, the most important white activist for black rights was William Lloyd Garrison, who was the most famous abolitionist in the years before the Civil War.
Perhaps the most important white person in the 1960s Civil Rights Movement was President Lyndon Johnson. While he wasn't an activist, it was he who pushed the Civil Rights Act of 1964 through Congress.
Andrew Goodman and Michael Schwerner were two of three civil right workers who disappeared in 1964. Their disappearance and subsequent murders were the topic of the 1988 film Mississippi Burning. James Chaney was the third activist that was also killed