NEW YORK — Some 60 years ago, a doctor in Baltimore removed cancer cells from a poor black patient named Henrietta Lacks without her knowledge or consent. Those cells eventually helped lead to a multitude of medical treatments and laid the groundwork for the multibillion-dollar biotech industry.

It’s a saga made famous by the 2010 bestseller “The Immortal Life of Henrietta Lacks.”

Now, for the first time, the Lacks family has been given a say over some research involving her cells....