As a candidate for Colorado governor, Walker Stapleton has touted a distinguished biography: two terms as state treasurer, a business degree from Harvard and a long family history of public service.

But there is one aspect of his family’s past that Mr. Stapleton has largely avoided mentioning: His great-grandfather, Benjamin Stapleton, a five-time mayor of this city, was also a powerful member of the Ku Klux Klan, a bespectacled former judge who helped the group seize control of Colorado government in what is now considered one of the state’s darkest periods.

Three years after a massacre at a black church in Charleston spurred a wave of American institutions to reconsider memorials to the Confederacy, Colorado is having its own reckoning. Here, the debate is not about Civil War statues, but about whether to rebrand institutions that have come to bear the Stapleton name: a well-to-do neighborhood, several schools, many businesses.

And now Walker Stapleton, the candidate for governor, finds himself in the difficult position of attempting to lead voters into the future just as the state grapples with his family’s past.