If you want to alter the widget, then you can rename it and put it as an additional widget in your functions file.
You will need to rename it in several places.
If you want to work on the core version and get it working, then if you want to post a revised core version (and any associated css) here, then I’ll quite happily work out the rename areas so you can then post it in your functions file – as long as your version is freely available for others (and me) to use.