(I'm actually asking this even though I intend to supply the answer, so this is more by way of providing a tip.)
If you create an emacsclient launcher by copying a .desktop file (e.g. the Emacs one), it does not behave in the standard Unity launcher manner. It always launches a new window; it doesn't recognize that existing Emacsclient windows exist to jump to them. How can I make it behave?
As background, this is a very useful usage mode for Emacs, where you run it as a daemon:
emacs --daemon, and then you summon an emacs window with
emacsclient. A good way to do this for habitual emacs users is to put
emacs --daemon into your Startup Applications.