WASHINGTON — When Theodore Roosevelt hired an architecture firm to renovate the dark, cramped White House in 1902, much of the structural interior was thrown into piles of trash on the lawn outside: 1600 Pennsylvania Avenue had become a construction site. Wood, plaster, glass and even old curtains were strewn about, leaving the once stately grounds in a heap of “dirt and confusion,” as an article in The New York Times put it.

Under orders from Roosevelt, the firm worked quickly to restore the site and complete renovations, without taking time to preserve or repurpose the wood and other materials that had surrounded every president since the building’s reconstruction after the War of 1812. A good chunk of history was carted away and burned.

But a single structural item from the White House of Abraham Lincoln and John Quincy Adams was preserved. Joseph Williamson Jr., a student at Georgetown law school, pulled a 14-by-30-inch piece of ornamental wood from one of those trash piles, with the help of his brother and a friend who was working on the site. Mr. Williamson took it home with him, writing “from old White House. Oct. 15, 1902” on the back in pencil.

Theodore Roosevelt’s trash became Mr. Williamson’s lifelong treasure. The piece of wood, called a plinth, resurfaced when one of Mr. Williamson’s distant relatives contacted the White House Historical Association about a family heirloom. The piece is now being put up for auction after Mr. Williamson’s family members decided to sell it after the loss of two farms that had been in the family for generations.