It's one of the quirks of history that the mediaeval Holy Roman Empire had very little territorial overlap with the old Roman Empire. What's the most plausible way that the old Roman territories of western Europe (France, Italy, Spain) could become part of the HRE, and what effect would this have on the development (social, cultural, political, technological) of Europe?
(The Emperor's actual control over these lands can be purely nominal if you want, just as long as the rulers here do homage of some sort to the Emperor.)