Drop-down menu with wp_dropdown_pages

First of all, i think you got an error in your arguments. It should be 'show_option_none' => 'Select Page', not 'show_option_none=Select Page'.

Secondly: Shortcodes are things that are replaced within the content of the post/page/whatever. This means that the content of the Shortcode has to be returned. You directly echo them, which leads to the error message in the admin.

Change your function to return the Code instead of echoing it. The wp_dropdown_pages function has a handy argument for this: ‘echo’.

function dropdownbox_autosubmit($atts = array()) {
    if(isset($atts['child_of'])){
         $childof = (int)$atts['child_of'];
    } else {
         $childof = 42;
    }
    $my_dropdown = wp_dropdown_pages(array('child_of' => $childof,'show_option_none'=> 'Select Page','echo' => false));
    $my_dropdown.='<script type="text/javascript">';
    $my_dropdown.='    var pageDropdown = document.getElementById("page_id");';
    $my_dropdown.='    pageDropdown.onchange = onPageSelect;';
    $my_dropdown.='    function onPageSelect() {';
    $my_dropdown.='        if ( pageDropdown.options[pageDropdown.selectedIndex].value > 0 ) {';
    $my_dropdown.='            location.href="https://wordpress.stackexchange.com/questions/363387/.get_home_url()."?p="+pageDropdown.options[pageDropdown.selectedIndex].value;';
    $my_dropdown.='         }';
    $my_dropdown.='     }';
    $my_dropdown.='</script>';

    return $my_dropdown;
}
add_shortcode( 'call_dropdownbox_autosubmit', 'dropdownbox_autosubmit' );

You can continue to use the shortcode like this: [call_dropdownbox_autosubmit]. I also made a little change so that you can specify of which page the children in the dropdown should be. You can use the child_of Attribute like this:

[call_dropdownbox_autosubmit childof=42]

Happy Coding!