Note this uses PyQt4. It was uploaded only for reference.

rename_shapes.py

# -*- coding: utf-8 -*-
# ***************************************************************************
#
# Code borrowed from TableManager
#
# Copyright (C) 2008 Borys Jurgiel
# Adapted by Germán Carrillo to rename Shapefiles (2015)
#
# ***************************************************************************
# *                                                                         *
# *   This program is free software; you can redistribute it and/or modify  *
# *   it under the terms of the GNU General Public License as published by  *
# *   the Free Software Foundation; either version 2 of the License, or     *
# *   (at your option) any later version.                                   *
# *                                                                         *
# ***************************************************************************

from qgis.core import *
from qgis.gui import *
from PyQt4.QtCore import *
from PyQt4.QtGui import *
import sys
DEBUGMODE=True
########## CLASS TableManager ##############################

class RenameShapefiles():
  def __init__(self, iface, layer):
    self.iface = iface
    self.layer = None
    if type(layer) is None:
      print "This doesn't seem to be a valid layer..."
      return
    if not layer.type() == QgsMapLayer.VectorLayer:
      print layer.name() + " is not a vector layer..."
      return
    self.layer = layer
    self.provider = self.layer.dataProvider()
    self.fields = self.readFields( self.provider.fields() )
    if not self.provider.storageType() == 'ESRI Shapefile': # Is provider saveable?
      print self.layer.name() + " is not a Shapefile layer..."
      return
    self.readData()

  def readFields(self, providerFields): # Populates the self.fields dictionary with providerFields
    fieldsDict = {}
    i=0
    for field in providerFields:
        fieldsDict.update({i:field})
        i+=1
    return fieldsDict

  def formatFieldType(self, field):
      if field.type() == QVariant.Date:
          return field.typeName()
      elif field.type() == QVariant.Double:
          return "%s(%d,%d)" % (field.typeName(), field.length(), field.precision())
      else:
          return "%s(%d)" % (field.typeName(), field.length())

  def readData(self): # Reads data from the 'provider' QgsDataProvider into the 'data' list [[column1] [column2] [column3]...]
    fields = self.fields
    self.data = []
    for i in range(len(fields)):
      self.data += [[]]
    for feat in self.provider.getFeatures():
        attrs = feat.attributes()

        for i in range(len(attrs)):
            self.data[i] += [attrs[i]]

  def doSave(self): # Called when appropriate button was pressed
    # I believe the procedure below is as much safe as possible.
    if not self.layer:
      return
    layerName = self.layer.name()
    srcName = layerName
    encoding = self.provider.encoding()
    tmpDir = QDir.tempPath()
    srcPath = self.provider.dataSourceUri().split('|')[0]
    srcOrigPath = QFileInfo(srcPath).path() + '/' + QFileInfo(srcPath).baseName()
    originalName = QFileInfo(srcPath).baseName()
    srcPath = QFileInfo(srcPath).path() + '/' + layerName
    if originalName == layerName:
      print "Layer "+ layerName+" has nothing to rename..."
      return
    # without this one line code in win xp, srcName return something wrong,see below (patch from Volkan Kepoglu - thanks!)
    srcPath = srcPath.replace("\\","/")
    srcOrigPath = srcOrigPath.replace("\\","/")

    # write the layer to the temporary directory
    if self.writeToFile(tmpDir+'/'+srcName+'.shp', encoding) != 0:
      QMessageBox.warning(self, 'Table Manager', "Failed saving the changes to the temporary directory: "+tmpDir+" !\nThe layer won't be changed, please use the Save As button.")
      QgsVectorFileWriter.deleteShapeFile(tmpDir+'/'+srcName+'.shp')
      return
    # try to remove the old .dbf~ backup
    QFile(srcOrigPath+'.dbf~').remove()

    QFile.remove( tmpDir+'/'+srcName+'.qml' )
    self.layer.saveNamedStyle( tmpDir+'/'+srcName+'.qml' )
    QgsMapLayerRegistry.instance().removeMapLayers([self.layer.id()])
    
    #Rename files
    QFile(srcOrigPath+'.shx').rename(srcPath+'.shx')
    QFile(srcOrigPath+'.prj').rename(srcPath+'.prj')
    QFile(srcOrigPath+'.qpj').rename(srcPath+'.qpj')
    QFile(srcOrigPath+'.shp').rename(srcPath+'.shp')
    
    ###########
    
    # rename the oryginal .dbf file to .dbf~
    if not QFile(srcOrigPath+'.dbf').rename(srcPath+'.dbf~'):
      QMessageBox.warning(self, 'Table Manager', 'Failed backuping the old table to '+srcPath+'.dbf~\nThe layer won\'t be changed, please use the Save As button.')
      # we don't return now because layer has to be reloaded'''
    # copy the .dbf from the temp directory to the target location
    elif QFile(tmpDir+'/'+srcName+'.dbf').copy(srcPath+'.dbf'):
      # dbf file copied. Copy also cpg if present
      pass
    else:
      # something went wrong with copying the dbf file. Restore the dbf~ file
      if not QFile(srcPath+'.dbf~').rename(srcPath+'.dbf'):
        QMessageBox.warning(self, 'Table Manager', 'WARNING! I can neither save the new {0}.dbf file\nnor restore it from the '+srcName+'.dbf~ backup.\nPlease check it manually!')
        QgsVectorFileWriter.deleteShapeFile(tmpDir+'/'+srcName+'.shp')
        return
      # dbf~ file restored :
      QMessageBox.warning(self, 'Table Manager', "Failed saving the changes to "+srcPath+".dbf\nThe layer will not be changed, please use the Save As button.")
      '''
      QgsVectorFileWriter.deleteShapeFile(tmpDir+'/'+srcName+'.shp')
      return
      we don't return now because layer has to be reloaded
      '''
    # clean the temp directory
    QgsVectorFileWriter.deleteShapeFile(tmpDir+'/'+srcName+'.shp')
    # create the new layer
    '''layerName = self.layer.name()'''
    newLayer = QgsVectorLayer(srcPath+'.shp', layerName, 'ogr')
    if not newLayer.isValid():
      QMessageBox.warning(self, 'Table Manager', "WARNING! The changes seem to be commited, but I can't reload the layer!\nPlease check it out!\nThe old table is backuped as "+srcName+".dbf~.")
      return
    # copy the style (it's possible only if the clasyfying field was not renamed!)
    '''#newLayer.copySymbologySettings(self.layer)'''
    resp = newLayer.loadNamedStyle( tmpDir+'/'+srcName+'.qml' )
    if not resp[1]:
      QMessageBox.warning(self, 'Table Manager', "This layer will be reloaded without its previous style (loading style failed)")
    QFile.remove( tmpDir+'/'+srcName+'.qml' )
    # set encoding
    newLayer.setProviderEncoding( encoding )
    # reload the layer
    QgsMapLayerRegistry.instance().addMapLayers([newLayer])
    # point the self.layer to the new one.
    self.layer = newLayer
    self.provider = self.layer.dataProvider()
    self.fields = self.readFields( self.provider.fields() )



  def writeToFile(self, filePath, encoding, driverName = 'ESRI Shapefile'): # write data to a shapefile
    if QFile(filePath).exists():
      try:
        QgsVectorFileWriter.deleteShapeFile(filePath)
      except:
        QMessageBox.warning(self, 'Table Manager', 'Cannot overwrite an existing shapefile.\nPlease remove it manually.')
        return 1
    # create destination layer
    fields = QgsFields()
    keys = self.fields.keys()
    keys.sort()
    for key in keys:
        fields.append(self.fields[key])
    if driverName == 'ESRI Shapefile':
        #lco = ["ENCODING="+encoding , "SHAPE_ENCODING="+encoding]
        lco = ["ENCODING="+encoding]
        settings = QSettings()
    else:
        lco = []
    writer = QgsVectorFileWriter(filePath, encoding, fields, self.provider.geometryType(), self.provider.crs(), driverName, [], lco)
    if writer.hasError():
      QMessageBox.warning(self, 'Table Manager', "Error creating file. The errror message was:" + "<br/>" + writer.errorMessage() )
      return 2
    geom = QgsGeometry()
    n = 0
    for feat in self.provider.getFeatures():
        geom = feat.geometry()
        outFeat = QgsFeature()
        if geom:
            outFeat.setGeometry(geom)
        attrs = []
        for i in range(len(self.fields)):
            try:
                attrs += [self.data[i][n]]
            except:
                attrs += ['']
        outFeat.initAttributes(len(attrs))
        outFeat.setAttributes(attrs)
        writer.addFeature(outFeat)
        n += 1

    del writer
    return 0