Can I register a new thumbnail size and have it be an option in “Add Media”?

Yes you can.

You have to use wordpress Settings API (tutorial) to add new options/fields into such pages.

Also add add_image_size function to register new size so wordpress create new image file according to given size.

As per your requirement you need 50×50 so I added icon_size field only you can change this as per your requirement.

Happy coding!

To show into media section

add_action('admin_init', 'stack_initialize_theme_options');
function stack_initialize_theme_options() {

    add_settings_section(
        'media_settings_section',
        'Custom Image sizes',
        'stack_media_options_callback',
        'media'
    );

    add_settings_field(
        'icon_size',
        'Icon size',
        'stack_toggle_size_callback',
        'media',
        'media_settings_section'
    );

    register_setting(
        'media',
        'icon_size'
    );
}

function stack_media_options_callback() {
    echo '';
}

function stack_toggle_size_callback($args) {

     $size = get_option('icon_size') ? : 50;

     $html="<label for="icon_size">Width</label>";
     $html .= '<input name="icon_size" type="number" step="1" min="0" id="icon_size" value="'.$size.'" class="small-text">';

     echo $html;
}

---

Register new image size

if ( function_exists( 'add_image_size' ) ) {
    add_image_size( 'icon_size', 50, 50, true );
}