For ever layer, set every attribute field's value to the base file name of the layer. Useful for later aggregating by field value.

set field value to layer name

import arcpy
from arcpy import env

env.workspace = "E:/master_regions"
files = arcpy.ListFiles("*.dbf")
for file in files:
     rows = arcpy.UpdateCursor(file)
     for row in rows:
          row.NAME = file.split('.')[0]
          rows.updateRow(row)