Finally I found a solution! And it is a really strange behaviour.
First of all you need to specify setDomStorageEnabled(true) on your webview. Otherwise the DOM doesn’t work. I wonder why no tutorial gave a hint about. But ok.
myview.getSettings().setDomStorageEnabled(true);
After this I ended up in a white blank page with only the value I set. The strange thing is, that javascript:document.getElementById('myfield').value="aaa";
returns a value. Namely the one I set. So a new blank page was created that only contains the string “aaa”.
I solved it by modifying the javascript to throw away the return result:
javascript:var x = document.getElementById('myfield').value="aaa";
And voilá. It is working.