Josh23french

functions.php gets called: it adds the custom action and removes nothing.
404.php gets called: it adds the default action and does the action

The custom text is shown, then the default, right?

You should put the default action in functions.php and the custom one in 404.php.

So:

# functions.php

function set_aisis_404_default_err_message(){
     _e('cannot find what you want');
}
add_action('aisis_404_err_message_banner', 'set_aisis_404_default_err_message' );

# 404.php

function set_new_aisis_404_err_message(){
    _e('booboo occured');
}

add_action('aisis_404_err_message_banner','set_new_aisis_404_err_message');
remove_action('aisis_404_err_message_banner', 'set_aisis_404_default_err_message' );

After that point in 404.php, whenever you call do_action('aisis_404_err_message_banner'); you’ll get what you want.

Where exactly do you add the default message? It seems as if you add it after your call to remove it.

WordPress will work perfectly fine without any external connections.