You can change this line:
'orderby' => 'post__in',
to this:
'orderby' => 'rand',
Instead of showing the images in the same order as they were passed to 'post__in', they will show in random order each time the page is loaded.
Keep in mind that if you update the child theme, your change will be overwritten.
EDIT: The list of “pre selected images” is passed through the shortcode on the page where you want to display the gallery. If you don’t want to pass a list of images, you could try commenting out this line (few lines above where you changed 'orderby'):
// 'post__in'=> explode( ',', $images ),
This will ignore any list of images you pass in the shortcode, and include all images on the site. In this case you may also want to adjust the following line:
'numberposts' => -1
The -1 is for unlimited, but you could change it to something that makes sense for the gallery page, like 20 (or whatever number you want).
For additional parameters, such as excluding images, see the Codex.