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 and export each to a different 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 and export each to its own file
# based on its name and by selection
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() accepts. For example, you can include normals with use_normals=True.