Child entity cloning has to be a part of the parent cloning process, and the method of copying all the values from the originals is identical to what's done with the parent. You may use Javascript at the client to do this, using Web Services calls via SOAP. To duplicate the children, you would obviously have to Query for them, which you may find
some code helpful with . Javascript doesn't run at the server side, so you'd instead be looking at a Plugin or custom Workflow activity to process it.
Javascript SOAP functions for this would at least be able to report the new record's id, and you could use the solution here:
http://social.microsoft.com/Forums/en-US/crmdevelopment/thread/486a2c77-8e2e-42f4-9622-8583fd415b82 to display it alongside your original record.
Dave Berry