I think what you want is to launch the Byobu desktop application. You can find that in the Ubuntu application launcher, by simply typing Byobu. You can lock it to the launcher, and the icon will always stay there, on the left.
Alternatively, most Linux terminals can be set to run a particular command or application at launch.
For instance, in Gnome Terminal, simply click Edit -> Profile Preferences, select the Title and Command tab, check the Run a custom command instead of my shell box, and enter /usr/bin/byobu.
Full disclosure: I am the author and maintainer of Byobu.