The Debian desktop is pretty much Gnome with some Debian wallpaper. The default Main Menu icon is the Gnome foot.
To replace the Gnome foot with the Debian Swirl, replace the start-here.svg or start-here.png image with an appropriate Debian Swirl image. (Other Gnome foot icons in the directory are just shortcuts to those images and will change automatically.)
/usr/share/icons/gnome/scalable/placesThen update the icon cache:
# gtk-update-icon-cache -f '/usr/share/icons/For a user-installed theme, the location and the image filename will be different:
/home/user/.icons/icon theme/scalable/placesA similar icon cache update will be required:
$ gtk-update-icon-cache -f '/home/Or simply delete the icon-theme.cache file from the icon theme directory and it will be rebuilt with the new icon.
Credit goes here and here.
Update : Edited the gtk-update-icon-cache commands following these instructions.