While Attorney General Herbert Brownell certainly was the leading advocate for civil rights in the cabinet (his original version of the bill before it was watered down in the Senate was actually far closer to the one passed in 1964 than what ended up passing), Nixon was the one who stood for most of the White House's efforts to actually get the bill through. In all probability, in particular when you consider his later Southern Strategy, Nixon was far more interested in the political harvest the Republicans would reap if they were seen as the party who were the force behind the new Civil Rights Bill. Though African-Americans had (and since always have) voted for the Democrats in much higher numbers than for the Republicans since Franklin D. Roosevelt's Presidency (prior to that they had staunchly backed the Party of Lincoln), and it looked very unlikely that the Republicans would ever again win a majority of the black vote, that didn't really matter. During Eisenhower's presidency, around 30% of African-Americans voted Republican, and Nixon calculated that if he could get that figure up to 35 or even 40%, that would greatly benefit the Republicans in a number of swing states, like New York, Wisconsin, Ohio, California, etc.
Meanwhile, throughout the battle for the Civil Rights Act of 1957, Eisenhower was for the most part somewhere between uninterested and apathetic to the whole enterprise. Indeed, Eisenhower would on occasion, during informal drinks and dinners, try to persuade his interlocators that, you know, those Southerners like Russell and Eastland and Thurmond, they weren't as bad as people seemed to think. They were reasonable. They weren't racist. All they wanted was that their sweet, innocent little girls shouldn't be forced to sit next to 150 pound ██████ in school.