PHILADELPHIA — Visitors to the Penn Museum might never see the red clay tablet. Little bigger than the palm of a hand, it sits on a metal cart in a back room.

Covered with indented rows of tiny characters, the Sumerian tablet dates from about 2700 B.C., and it is the world’s first known written account of the biblical flood. When not on its cart for visitors to see and handle, it is stored, like many of the museum’s one million other objects, in stacks of metal drawers accessible only to academics and other researchers.

The museum, formally called the University of Pennsylvania Museum of Archaeology and Anthropology, has over the years become an internationally renowned treasure trove for scholars researching ancient civilizations. Now to mark its 125th anniversary, and its founding on Dec. 6, 1887, the museum is undertaking an ambitious effort to become more accessible to the public....