You need to do this in a two-stage process using the Vector > Analysis Tools > Mean Coordinates tool in the second step. This tool will return the mean coordinates for sets of point within a layer if they have a unique "ID" field. So, if you have a polygon layer which defines your areas, do a spatial join (Vector > Data Management > Join attributes by location) to attribute your points with the polygon (area) id and then use the Mean coordinates tool, specifying the new field.
You may already have unique "ID"s or a polygon layer but if you do not you could create a fishnet (Vector > Research Tools > Vector Grid) to lay over the points as one method to define the areas. Alternatively, if your areas are defined by a raster layer, then you could use the Point sampling plugin to attribute the points based on your raster. Or, if you want to attribute them by location without a vector or raster layer, you will need to create an algorithm to give a new attribute based on the coordinates of each point (possibly using some modulus-based approach).