SPRINGFIELD, Ill.—The War of Santa Anna’s Leg has raged for 169 years. It started when the Mexican general abandoned his wood, cork and leather prosthetic in a carriage, and resumed most recently when a group of classmates from Texas took a 20-hour pilgrimage by bus to a small museum in a bid to get it back.

“For any other person it’s just a prosthetic leg, you know?” said Andre Grajeda, a 19-year-old sophomore history major at St. Mary’s University, San Antonio. “For me, it was kind of majestic, and it was magical.”

The...