I released a script a few days ago that allows you to have the mesh names matched to the object name on all selected objects. This script is exactly the same, but does the opposite. For every selected mesh, it replaces the objects name with the mesh name.
Install like any other add-on.
Select objects -> Space -> Mesh to object name
cypher2012 . Thanks! Very useful, I’, really bugged by how Blender does not rename both automatically. Really makes a mess when exporting OBJ. Oh well, community to the rescue!
mkbreuer . Thanks on adding it to the Outliner File Menu, much appreciated!!