Ten days ago, Lisa Cook was one of seven members of the Federal Reserve’s board of governors, playing an important but hardly leading role in the central bank’s debate over the path of interest rates.

Today, the future of the Fed and whether it will continue to operate as an independent institution or become subject to the whims of the White House rests largely on her shoulders.

Ms. Cook finds herself in that position because of President Trump’s decision on Monday to seek her ouster, and her own decision on Thursday to file a lawsuit challenging her attempted dismissal. Those two actions set the stage for a landmark legal battle, one that is bound for the Supreme Court, over the president’s explicit attempts to take control of the central bank.

“Governor Cook has been thrust into a role she did not seek and doubtless would prefer to shed — suddenly being cast in a larger-than-life struggle to defend an institution that has helped foster the economic and financial success of the U.S. over the post-World War II period,” said David Wilcox, who is a senior fellow at the Peterson Institute for International Economics and a former leader of the Fed’s research and statistics division.

On Friday, a federal judge in Washington will hold the first hearing in the case, which will focus on Ms. Cook’s request for a temporary restraining order. If the court grants the order, it will allow her to continue serving on the Fed’s board while she contests her firing.

Ms. Cook was not expected to be in this position. Mr. Trump has made no secret of his desire for lower interest rates, or of his anger at central bank officials for refusing to deliver them. But for months Mr. Trump had focused his ire on Jerome H. Powell, the Fed chair, repeatedly threatening to fire him and, at one point, even waving around a letter that purported to do so.