Since you are on Trisquel 8, you can execute: $ gksu -u webbrowser abrowser
Does it work? Notie that, as far as I understand, 'gksu' is now considered insecure and 'pkexec', in Polkit is supposed to replace it. What I suggested you is essentially an adaptation of https://askubuntu.com/questions/287845/how-to-configure-pkexec/332847#332847 and I do not understand why that does not work.
In your initial post, you wrote "I understand I need to do something with sudo". sudo may indeed be another solution. You can execute 'sudo -Hu webbrowser abrowser' in a terminal. However, I believe you do not want to have to open a terminal to launch your Web browser. For a graphical way to input the password:
$ SUDO_ASKPASS=/usr/bin/ssh-askpass sudo -AHu webbrowser abrowser In a launcher, the command would be: sh -c 'SUDO_ASKPASS=/usr/bin/ssh-askpass sudo -AHu webbrowser abrowser'
