I had this problem, couldn't find a solution online, and did the following to get it working.
First, one line at a time, paste these commands into terminal and run.
sudo add-apt-repository ppa:danielrichter2007/grub-customizer
The repository had to be added before I could do the next step:
sudo apt-get update
sudo apt-get install grub-customizer
After these steps, Grub Customizer still didn't show up in System Tools menu, which probably means the app is not installed.
Now, I'm not sure what is necessary to do and what is not, but doing the following got Grub Customizer installed and running. (Again, one step at a time.)
sudo apt-get upgrade grub-customizer
sudo apt-get build-dep grub-customizer
Finally, I went into synaptic, typed "grub" and grub-customizer finally showed up. I had to install it from there. Why it didn't show up in synaptic once the repository was added in the first place, I don't know. But, now it shows up in the application menu, launches, and functions at least on the first tab.