How do I change the desktop name on the Unity panel?

There's a much quicker way than doug's method, by abusing the gettext translation system. Assuming you are using an English locale:

#Make a temporary text file using vim/gedit/cat/whatever
cat > /tmp/foo.po
msgid "Ubuntu Desktop"
msgstr "A Badger With A Gun"
^D

cd /usr/share/locale/en/LC_MESSAGES
sudo msgfmt -o unity.mo /tmp/foo.po

Then, either log out and back in, or just execute:

unity --replace

You would likely need to build the unity or unity-2d source. Currently, in unity-5.12, it would be found in /plugins/unityshell/src/PanelMenuView.cpp, line 78:

 _desktop_name(_("Ubuntu Desktop"))

In unity-2d, unity-2d 5.12: /panel/applets/appname/appnameapplet.cpp, line 369:

d->m_label->setText(u2dTr("Ubuntu Desktop"));

It appeared that, The Text on top-left corner uses the value from the file /usr/lib/os-release file. So, We need to edit the value for NAME key on that file only.

1. Open with nano

   sudo nano /usr/lib/os-release
   

2. Change the value of NAME from the first line. I changed it to L Buntu just for testing. The file's content after change

   NAME="L Buntu"
   VERSION="16.04.1 LTS (Xenial Xerus)"
   ID=ubuntu
   ID_LIKE=debian
   PRETTY_NAME="Ubuntu 16.04.1 LTS"
   VERSION_ID="16.04"
   HOME_URL="http://www.ubuntu.com/"
   SUPPORT_URL="http://help.ubuntu.com/"
   BUG_REPORT_URL="http://bugs.launchpad.net/ubuntu/"
   UBUNTU_CODENAME=xenial
   

3. Logout and Login again to see the change.

   Here is a tiny screenshot after change.

   

Tested on 16.04.

This is a bit unusual and may be reverted if you update Unity, but this should be possible.

1. Open GEdit (or any other text editor) and insert the following:

   msgid "Ubuntu Desktop"
   msgstr "Mac OS X"
   

2. Save the file wherever you'd like with the .po extension. I'll refer to it as ~/desktop.po.

3. Open up the terminal and execute the following commands:

   cd /usr/share/locale/en/LC_MESSAGES
   sudo msgfmt -o unity.mo ~/desktop.po
   

4. Log out and log back in and the text should have changed.

This is taking advantage of the translation system in Ubuntu to make it consider the English translation of "Ubuntu Desktop" to be "Mac OS X".