There doesn't exist a way to do this in the options of the obj importer it seems so what you could do is run a script that loops over the items in your scene, check if it is as mesh and export the current selection to its own file relative to the path your blend file is saved.

    import bpy
    
    # get the path where the blend file is located
    path = bpy.path.abspath('//')
    
    # loop through all the objects in the scene
    for ob in bpy.context.scene.objects:
        # make the current object active and select it
        bpy.context.scene.objects.active = bpy.data.objects[ob.name]
        ob.select = True
        
        # make sure that we only export meshes
        if ob.type == 'MESH':
            # export the currently selected object to its own file based on its name
            bpy.ops.export_scene.obj(filepath=str(path + ob.name + '.obj'), use_selection=True)
        # deselect the object and move on to another if any more are left
        ob.select = False

To further tweak your export options, see the parameters that [`bpy.ops.export_scene_obj()`](http://www.blender.org/documentation/blender_python_api_2_69_4/bpy.ops.export_scene.html#bpy.ops.export_scene.obj) accepts. For example, you can include normals with `use_normals=True`, change forward axis with `axis_forward` etc.

There are more stringent checks that could be made but for simple use cases and if used properly, this should suffice.

Also see http://blender.stackexchange.com/questions/5064/batch-import-wavefront-obj/5065#5065.