/usr/share/gdm/greeter-config/20_debian

You will notice that this file is in fact a symbolic link to /etc/gdm3/greeter.gconf-defaults which happens to get overwritten with the default file on every restart. However it is 20_debian that gets read by gdm3 so you can replace this symbolic link with the file with your changes. To do so run the following:

sudo rm /usr/share/gdm/greeter-config/20_debian
sudo cp /etc/gdm3/greeter.gconf-defaults /usr/share/gdm/greeter-config/20_debian

This will replace the symbolic link with the default file and you can then run sudo gedit /usr/share/gdm/greeter-config/20_debian to make your changes.

The default file will look like this:

# Use a specific background
/desktop/gnome/background/picture_filename   /usr/share/backgrounds/linuxmint/Talento-1.jpg
/desktop/gnome/background/picture_options   stretched

# Theming options
/desktop/gnome/interface/gtk_theme      Shiki-Wise
#/desktop/gnome/interface/icon_theme      gnome-wise

# Greeter options
#/apps/gdm/simple-greeter/logo_icon_name    debian-swirl

# Some other possible options
#/apps/gdm/simple-greeter/banner_message_enable      true
#/apps/gdm/simple-greeter/banner_message_text      Welcome
#/apps/gdm/simple-greeter/disable_restart_buttons   false
#/apps/gdm/simple-greeter/disable_user_list      false

# The lower panel doesn't work with the compositor
/apps/metacity/general/compositing_manager      false

Here you can change the background picture, gtk theme and icons used. The themes, icons and backgrounds will need to be available to the gdm user and therefore need to be in /usr/share/(backgrounds/themes/icons).

Once your changes have been made run the following command:

sudo dpkg-reconfigure gdm3