The Big "Where?" Question
It's nearly impossible to establish Jebel al Lawz as the location of Mt Sinai based solely on physical remains today. The petroglyphs at the site of Jebel al Lawz are almost universally acknowledged to be from the Nabataean period. I've watched one of the several videos put out about Jebel al Lawz (this particular one by Bob Cornuke - which at least didn't have the grainy "bigfoot" footage of some previous videos), located the area on Google Maps to examine a few things for myself, and spent a while considering it seriously. I don't think the physical evidence is conclusive as to that particular mountain. However, the larger debate inside (calmer and less hyped) academia, in historical and anthropological circles, is more general: does the Exodus text describe a crossing at the modern Gulf of Suez or Aqaba? The debate continues to rage there.
Part of that debate ties into logistics (how and where could so many people cross?) and also the identification of the Yam Suph (the designation "Red Sea" is a mistranslation at best where it is rather "Sea of Reeds", and is entirely misleading since there is no clear relationship between the Sea of Reeds and the Red Sea). See the brief Wikipedia entry here on that topic.
I am among those who lean toward an Arabian crossing, mainly because of the location of ancient Midian. There is actually someone who wrote a whole Ph.D. Dissertation arguing for Jebel al Lawz as the location of Mt. Sinai, which has some useful examination of the textual evidence (I'm not endorsing it, but it demonstrates the kind of texts one must deal with). But there is also a whole website against Jebel al Lawz being Mt Sinai. That debate is too specific (trying to name the particular mountain), IMO, and should focus more on whether the crossing was into Arabia or not generally speaking. Some people also appeal to the Apostle Paul's reference in Galatians 4:25 of "Mt Sinai in Arabia", though it is disputed how his ancient audience would have understood "Arabia" in that day.
Some general resources for weighing information about Midian
As with a lot of historical sleuthing work, often the less sensational sources that require more time and effort to sink your time into will pay off and allow you to piece evidence together yourself little by little. Knowing more about Arabia and also the Transjordan in the general period of the Exodus (which itself is subject to much chronological debate) I think will help you in this area if you want to investigate whether the crossing into the territory of Midian is a viable option.
A good scholarly volume to start with for the Transjordan might be Early Edom and Moab: The Beginning of the Iron Age in Southern Jordan (Sheffield Archaeological Monographs 7) edited by Piotr Beinkowski, especially Kenneth Kitchen's discussion of Egyptian excursions into the area (often no further south than Edom or Sier except to Timna) which testify to the presence of the pastoral nomadic peoples in those lands early on. I can't remember if that volume has information on the Arabian spice trade of the 12th century B.C. or not, but that's another topic to research to find out about the Midianites since their location makes it likely that they participated in that trade.
Also if you want to learn more about the Midianites I would suggest consulting several of Beno Rothenberg's volumes and excavation reports where he discusses "Midianite Ware" pottery that he uncovered at Timna, which later became known as "Qurayya Painted Ware" when a large volume of the same pottery was found in Qurayya in Arabia.
The Wikipedia article on Midian states:
Midianite pottery, also called Qurayyah Painted Ware (QPW), is found at numerous sites stretching from the southern Levant to NW Saudi Arabia, the Hejaz; Qurayyah in NW Saudi Arabia is thought to be its original location of manufacture. The pottery is bichrome / polychrome style and it dates as early as the 13th century BC; its many geometric, human, and animal motifs are painted in browns and dark reds on a pinkish-tan slip. "Midianite" pottery is found in its largest quantities at metallurgical sites in the southern Levant, especially Timna.
You may find a brief bibliography for Rothenberg below.
Rothenberg. B. 1969. The Egyptian temple of Timna.
Rothenberg, B. 1972. Timna, Valley of the Biblical Copper Mines. London.
Rothenberg, B. & Glass, J. 1983. The Midianite Pottery. In: J.F.A. Sawyer & D.J.A. Clines (eds), Midian, Moab and Edom (= Journal of the Study of the Old Testament, Suppl. 24, Sheffield): 65-124.
Rothenberg, B. 1987. Pharaonic Copper Mines in South Sinai. Iams 10/11: 1-7.
Rothenberg, B. 1988. The Egyptian Mining Temple at Timna. London.
Rothenberg, B. 1998. Who were the ‘Midianite’ copper miners of the Arabah ? About the ‘Midianite enigma’. In: Th. Rehren, A. Hauptmann & J. Muhly (eds), Metallurgica Antiqua, (= Der Anschnitt, Beiheft 8, Bochum): 197-212.
This article on Midianite Pottery is also packed with information that liberally draws upon Rothenberg's findings, and even briefly mentions the Mt. Sinai issue.
All that to say, you will have to do some research of your own for historical, archaeological, and geographical sources to allow you to make up your own mind, because no one has come to a consensus yet where the historical Mt. Sinai is.