There doesn't exist a way to do this in the options of the obj importer so what you could do is run a script that loops over the items in your scene, check if it is a 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('//')

    # deselect all objects
    bpy.ops.object.select_all(action='DESELECT')    

    # loop through all the objects in the scene
    scene = bpy.context.scene
    for ob in scene.objects:
        # make the current object active and select it
        scene.objects.active = ob
        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_release/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.

To have it export only selected objects, comment line `7` and change line `11`.

    # remove
    bpy.ops.object.select_all(action='DESELECT')
    # change to this where instead of the entire scene, we only look through selected objects
    for ob in bpy.context.selected_objects:

or keep line `11` and add another line

    for ob in scene.objects:
        # add the check to see if in selected objects here
        if ob in bpy.context.selected_objects:
            # adjust indentation
            ...
            ...