This whole "Holy Roman Empire"/"sacrum romanum imperium" thing as an actual name happened kinda dynamically over time anyway; I don't think people sat down and figured out "this is the best name to use"; it rather developed due to external and internal circumstances - sometime during the interregum?
But it didn't start out like that. Even during Otto Is time it was still called "regnum francorum orientalium", so basically East-Frankish Kingdom, and nobody cared. The "Emperors" occasionally used different titles titles, too; it's nowhere near as clear-cut as everyone just deciding to go "Kaiser" and be done with it.
Then there was the business with the Popes (and I presume other interested parties) calling it "regnum teutonicum", specifically to tell everyone that they had no business being in Italy (or anywhere else except for Germany). Giving it the more "global" "Roman" legitimacy is pretty much a logical counter-move in telling everyone that Italy, of course, is legitimate Imperial territory, so the Emperors insisted on "romanum imperium" (that and the historical prestige the title always had, of course). It's also why being a King wasn't enough - good enough to rule German territory, not good enough for the Italian parts.
So you had the Roman part down to legitimize your rule, and that left getting rid of the pesky (and costly) Papal influence, and so during the Staufer-reign the "Holy" part was added (or started to being added) - the Emperors didn't need the Pope to legitimize them anymore, their rule already was both secular and religious (of course that didn't work out quite that easily, but, well, it was worth an attempt, maybe).
It's pretty much one long history of propaganda attempts, I suppose.