“refer to seneh,the thorn bush, or it may be derived from the name of Sin, the ancient Semitic moon god. The word Horeb means 'the desolate place'” (Baker's Bible Atlas, 76).
The precise location of Mt. Sinai is much in dispute. Long-standing tradition places it near the southern tip of the peninsula between the Suez and Aqaba gulfs (the two northerly arms of the Red Sea).
Others propose a location east of Aqaba, in ancient Midian. Neither geopolitical location overtly contradicts the Biblical record. Both were “in Arabia” in Paul's time (Gal. 4.25); and both were “out of Egypt” at the Exodus (cf. Ex. 12.37, 39, 41, 51; 13.3, 18). Elijah took about 40 days getting there from Beersheba (about 250 miles away from either location). The Israelites covered a similar distance in two months (Ex. 19.1).
Some reject the Midian location on the basis of Exodus 18.27 and Numbers 10.29-30, which seem to suggest a distinction between the two areas. But the phrase, “his own land,” is far too vague to be definitive either way. Num. 10.31 indicates that Moses' Midianite relatives, who may have been mildly nomadic, were intimately familiar with living in the Sinai wilderness, and there are a few ambiguous connections between Midian and Sinai in Scripture (cf. Ex. 3.1; Acts 7.29-30).
At any rate, the peninsula location is the most received, and may be the most plausible topographically and logistically. And Judges 11.16 does seem to support a location west of the Aquaba gulf, as it appears that Aquaba was the eastern most limit of their wilderness journey, before turning north to Kadesh. This, however, also is not irresistible.