way to include own functions independent of theme

This is not quite simple (as copying and pasting into new functions.php :D), but you can create a plugin that will load this every time.

A (very) good and simple guide can be found here.

Addition by Original Poster (johannes)

I created a plugin for that:

1. I created a simple php file and put it into wp-content/plugins. It looks basically like this (i.e. it contains the plugin name, description, author etc. and the function I want to use in my site across all themes):

   <?php
   /**
    * Plugin Name: Function-Includer
    * Plugin URI: my URI
    * Description: my description
    * Version: 1.0
    * Author: johannes
    * Author URI: my URI
    */
   
   [... code of my function ...]
   
   ?>
   

2. Then I went into the WP admin backend, to the plugin page, where that plugin is listed as “Function-Includer” and activated it.

3. Done – the function is available everywhere in my site now.