Unfortunately, that is not possible directly right now. However, you can do it manually. For that you would regenerate the mesh with ToElementMesh and specify a BoundaryMarkerFunction and a PointMarkerFunction. So, the None needs to be replaced with a function that does what you want. (See the documentation of ToElementMesh in the options section to see how those are written)
m2 = ToElementMesh["Coordinates" -> mesh["Coordinates"],
"MeshElements" -> mesh["MeshElements"],
"BoundaryMarkerFunction" -> None, "PointMarkerFunction" -> None]