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 your blend file is located
path = bpy.path.abspath('//')
# iterate through your scene and export each object to a different file based on its name
for ob in bpy.context.scene.objects:
bpy.ops.export_scene.obj(filepath=str(path + ob.name + '.obj'))