I think that you can find a variety of answers to this question. My own answer would be that Washington was one of the first voices to discuss what it means to be Black in America after the Civil War. Washington's own exploration of this condition is something that still comprises the understanding of African- American studies today. Washington's discussion of the Black predicament in America, his own experience with White society, and his pointed discourse with W.E.B. DuBois is something that is highly relevant today, in the sense that there is still heavy and heated debate about how Blacks in America can understand their own predicament and conception as the "hyphenated American." I think that this is something to which Washington holds importance. It is here where I think Washington is important. He articulated a condition that is still experienced today. We might have more information, but the fundamental question as to how to deal with the issue of "differance," as Derrida would put it, is something that animated Washington's writing. From this, we are still struggling to understand the concept as Washington struggled.
He was a influential African-American of that time. He felt that Blacks first had to become economically independent before they could fight for equal civil rights given to the Whites.