It was two weeks before the liftoff of the Apollo 11 mission when Thomas Moser’s boss walked into his office at NASA and announced, “We’re putting a flag on the moon.”

Mr. Moser, then a 30-year-old mechanical engineer, was put in charge of designing a flag mechanism that could not only fit into the lunar module and survive the flight, but also make the flag appear to fly on the windless moon.

His solution involved two sections of a staff, a telescoping tube and a nylon flag bought at a local housing goods store (Sears, he thinks). But in order for the flag to fit the staff, its edges needed to be trimmed. “They were throwing it all in the trash,” Mr. Moser recalled of the remnants in a recent interview, “so I picked it up out of the trash can, mounted it and had Neil Armstrong sign it.”

Forty-two years later, Mr. Moser is auctioning off those flag remnants. The expected selling price: $100,000....