It was originally just called the Roman Empire. There wasn't an attempt to create something different from the Roman Empire of old when Pope Leo crowned Charlemangne. Charlemagne included the title of Augustus in his official titles and it was a serious diplomatic problem between them and the Byzantine Empire (which viewed itself as the Roman Empire).

It changed following the Investiture Crisis. For years the Ottonian and Salian dynasties had relied on the church to counter balance their unruly vassals. When they lost to the papacy they were put in a pretty weak position.

Part of trying to reclaim imperial power under Fredrick Barbarossa was adding "Holy" to Roman Empire in the 12th century. Which emphasized that the empire and, by extension emperor, were sacred. Giving him fire against a hostile papacy that wanted independence.

In the end, the imperial government lost ground and became a mostly Germanic and central European empire. Changes were made to negate the pope's influence in approving the emperor and the real power moved to the individual dominions. But the name stuck around.