What does Sinai mean? Is there a shoresh to the word? Har Sinai - Is Mount Horev the same place?

Though there are many midrashim on the question you raise, the Ibn Ezra (Shmot 3) and the Ramban (Devarim 1) explain- as does Pirkei R’ Eliezer (40)- that Sinai is named for the “sneh”- the burning bush that Moshe found there. The Ramban writes that the Sinai desert in its entirety was named for Mt. Sinai and that Horev was the location immediately adjacent to Mt. Sinai so that the mountain was also called Mt. Horev.