Question: How do we respond to the argument that there is no archaeological proof of the Exodus?
Answer: The Bible has nothing to fear from the spade. Archaeology overwhelmingly supports the biblical narratives, and not even one discovery has ever sullied the Bible's historical accuracy. The Exodus, however, is an archaeological anomaly, because in the 150-plus years of earnest digging-about in biblical lands, no one has turned up any significant artifacts along the assumed route of the wilderness wanderings. There are two main assumptions that underpin this challenge, the first being that ancient Egypt left no record of the Exodus and it therefore did not occur. Second, that two-million people with all their accouterments would have left some durable rubbish along the way, and that a significant portion of that rubbish is discoverable. Since you are asking how we should respond to the skeptic, I shall focus on some extra-biblical reasons why these objections should not be given the weight that they seem to carry. Following are five rebuttals.
First, consider the gentle footprint of the wilderness wanderings as compared to the heavy footprint of a permanent city. During the Exodus, the Israelites were essentially nomads. They moved lightly over the land and never permanently lived on it. They planted no crops. They built no barns. They lived in tents—hey, even God "lived" in a tent back then! So I must ask, what does one expect to find in the way of artifacts—especially after thousands of years, and on a path that escapes contemporary certainty? Not much in my opinion. Permanent settlements are where we find artifacts. It is the householder or the farmer who stays and accumulates life's hardware, not the nomad. A person who knows that he'll soon have to move everything he owns must consider every item to be a potential burden. The effect of the lightness is twofold: First, there were no permanent dwellings, the foundations of which are themselves artifacts, and second, there were no accumulations of the extra goods that are associated with permanent dwellings.
The Bible gives us a very clear record of God's supernatural intervention in this lightness. First, he provided a food that could be easily gathered, manna (Ex. 16:31). This in itself kept the Hebrews from ripping up the land. Second, he didn't even let their shoes wear out in 40 years! (Deut. 29:5) So, in a way they never even left a footprint. Third, he provided water at Horeb (Ex. 17:6) without the digging of any wells. Fourth, even Caleb's bodily energy was conserved during the wanderings. At 85 years of age he was still a bring-on-the-Anakim kind of guy upon entering the Promised Land (Joshua 14:10-12). In summary, the Exodus was a divine journey, and it is my opinion that God conserved the land for his own glory and for the sake of his people.
My second extra-biblical thought concerns the lack of documentary evidence in the Egyptian writings, because you would think that such a traumatic event as the Exodus would have been recorded on a wall somewhere. Well, you'd think this until you read their walls and documents. Pharaohs (and kings in general) did not record their defeats for all posterity. The records were more like, "I [name here] am god—and have totally vanquished every nation in the whole world." You will not see, "Those Hebrews really took it to us! First, there were these plagues—we had to let them go! Then their God destroyed a good part of my army by a miracle at the Red Sea. Boy, he sure beat the snot out of us!" We have solid evidence that Egypt expunged records. Scholars are aware of a ruling people called the Hyksos whose records the Egyptians have (almost) eliminated. There is, therefore, no reason to doubt that the Hebrew record might have also been expunged—that is, if it were ever recorded in the first place. I've found some interesting discussions on this and other Exodus related items at the following website. I suggest you visit it.
Third, both biblical and secular historians agree that the biblical place names and route descriptions of the Exodus have no necessary relationship to our contemporary notions of where those places or routes might be. The Sinai Peninsula is a big place! So, where would one look for an artifact? It is not unreasonable to assume that given the ambiguous routes, the vast expanse of the land, and the thousands of years that have lapsed, that there would be a lesser density of artifacts. For this reason alone, that wilderness is not a target-rich environment, I am not troubled by the lack of findings. It is reasonable not to have found evidence.
Fourth, the science of archeology has only been around for 150 years or so, and in that time only a tiny fraction of the biblical geography has been explored, and of the artifacts discovered in that tiny fraction of land only a tiny fraction have been analyzed. In terms of completeness then, archeology is virtually a total failure. Therefore, do not accept the notion that the failures of archeology are the failures of God...and yes, it does sound ridiculous when you undress it like that, but isn't that what they are asking us to do?
My fifth and final extra-biblical response involves some logic and some history. Logically, the purported lack of artifacts is an argument from silence, and an argument from silence is a categorically weak argument. Why so? Because a current lack of evidence is no proof that there will be a continual lack of evidence. This is often put as, "Absence of evidence is not evidence of absence." Time always vindicates the Bible. God is longsuffering with us and so we should be with him. Consider the following parallel problem.
The Hittites were a biblically significant people, but they were a people with no corroborating records. Archaeologists had been at their craft for decades with new discoveries at every turn, but they had never found a record of the Hittites. And because of this lack of discovery, people began to lose confidence in the Bible's veracity. But in 1906 archaeologists who were working in Turkey found sure evidence that the Hittites had indeed existed—and existed in a big way—just as the Bible described. In this we have our model. Do not allow archaeology, a discipline, which burns with the fuel of continual discovery, to ruin your weekend because it has not found something yet...and I'm a little worried that this might be a news-flash to some people.
You asked how should we respond to people who were impressed by this lack of evidence? I'd respond, "Do not let the Exodus send you to Hell!"—because it's just the Hittites all over again. Anyone who chose the lack of evidence concerning the Hittites, over the veracity of God's Word is now in hell. They do not get a do-over just because archeology had not yet progressed that far. In like manner, anyone who prefers the lack of secular evidence for the Exodus to the abundance of the Bible's many corroborated truths is in a wilderness of their own making...but these shall not likely reach the Promised Land. This is sad and unnecessary.
With this in view, I would suggest that all challengers perform a rewards-versus-risk analysis. A new discovery could be made today, and the secular world could reach a new Exodus consensus overnight! Skeptics should also realize that an argument from silence is a weak argument—and I would not want to dangle over hell by such a fine thread.