Gemara: It was called Sinai because from it hatred (sina) came down to the idol worshippers. And what is its [real] name? "Chorev." This argues with Rabbi Avahu, who said that its [real] name was Har Sinai and it was called Har Chorev because it was from it that destruction (churva) came to the idol worshippers.
Ein Ayah: The whole foundation of the improvement of the world, which will occur in the future and whose early stage already exists in the present, due to the great power of the acceptance of the Torah, was prepared by the previous state of destruction. Specifically, the world had been missing the basis for its existence because it was missing the divine light of the sanctity of the Torah, which would later give it light and adornment. From the midst of the world’s state of destruction, there developed a desired "edifice" of spirituality that came from the appearance of the Torah.
This would not have been possible if there were a structure in place, even if it was low and lowly, because there would still not be a possibility to have a new edifice appear. The situation, though, is that from the beginning of creation, had it not been for the power of Torah, the world would have been slated for destruction. It is only the light of Torah that gave the characteristic of being able to be built and to stand. After all, all of the special qualities related to the revelation of the Torah impact on the world and make it more adorned. This is based on the idea of building on top of the foundation of that which was previously missing. This was evident from the original name of the mountain that Hashem desired, so that He should give the Torah on it. That name, Chorev, hints at the state of destruction.
The other approach among Amora’im is that the real name of the mountain is Sinai, and that Chorev stands for that which happened on the mountain causing destruction to the idol worshippers. The world could have been "built" as a coarsely materialistic edifice. If that were the case, it would actually have been much worse than the fact that it was created in a manner that it is slated for destruction if left on its own. However, in the essence of Har Sinai and the revelation that occurred there, an internal hatred was embedded there. This serves as a strong protest against the way the world was built with a foundation connected to blood and metal, evil and foolishness. It would have been a tragedy if such a negative edifice would have continued.
Rather, Sinai was called Chorev because it brought destruction for idol worshippers. Even though they never became imbued with the powerful obligations of Har Sinai, what occurred there for Israel ensured that there would never be a quiet and stable edifice in a manner that contradicts the divine revelation of the mountain that Hashem desired would take place at Sinai with its sanctity. There needed to be some level of destruction, because of the world’s connection to lies and the essence of evil. Eventually, this enables there to be the revelation of a complete world, built on kindness (see Tehillim 89:3).