The Holy Roman Empire was Holy, was (technically) Roman, and was (technically) an Empire.
Hear me out. Starting with the most troubling cause, that "Roman" bit, let me go into some detail. The Catholic church was an organization that originated in the roman empire. After the roman empire split into the Eastern (Byzantine) and western sides, and the western half fell, the catholic church, being the central religious institution of the western roman empire at the time, was really all that was left of it. With the pope being the closest equivalent to a leader that this remnant of the original roman empire had. The pope also had control over the city of rome, making him, by default, the leader of what was once "Rome". Charlamagne, a the king that founded the carolingian empire, (Precursor to the HRE) was personally given the title of "Roman Emperor" by the pope, who, based on the fact that said pope owned Rome, and had control of a major "Roman" institution, would have a technical authority to grant such a title. As such, we can at least establish that the HRE was ruled by "Roman" emperors, thus making it roman.
Secondly, "Holy". This is an easy one. The catholic church was incredibly influential during this time period. Basically, if they said a place was holy, it for all intents and purposes was. The pope declaring an empire to be holy was, in that time period, the exact same thing as it being holy. Plus, even if you don't buy that, with holiness being a very subjective thing, variable depending on what your religious beliefs are, even if the HRE isn't holy to you, it probably was to someone at some point, which at least in my opinion justifies the name.
Finally, empire. Technically, the HRE was a confederation of kingdoms, and not a imperial dictatorship. However, we must note that the word Empire doesn't always have to refer to the same sort of thing we saw in the original Roman Empire. In the modern day and in the past, the U.S was often referred to as an empire for its imperialist tendencies and occasional covert annexation of islands. The word empire has also been used to describe corporate enterprises as well, but getting passed mere slang terms, let's look at the definition of the word itself.
"Empire: A political unit having an extensive territory or comprising a number of territories or nations and ruled by a single supreme authority."
Well, the HRE was one of the largest nations of europe in the days following the fall of the western roman empire, and, though it was a confederacy, all of the local monarchs did answer to the Holy Roman Emperor, so, in that case, there was a supreme authority.
I rest my case, your honor.
