Renaming attributes/fields in shapefile attribute table using QGIS?

Since 2.16, you can edit field names in the layer properties when editing is enabled:

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Renaming is possible using Table Manager plugin (http://plugins.qgis.org/plugins/tablemanager/)

Since QGIS 2.16 you no longer need a plugin to do this - you can double click a field name in a shapefile from the vector layer properties window and rename (set the layer as editable first)

if anyone is having any difficulty finding the Table manager button, for me it was located here: