I was reading through the wiki about no evidence of the exodus out of Egypt by the Israelites. What evidence would be used by historians of proof of the israelites? Writings, goods, campsites?
Oh boy!
Welcome, in part, to my dissertation.
This is a complex issue--and one that many disagree on. Let's start with a handful of helpful sources:
Norman Yoffee, Myths of the Archaic State
Avraham Faust, Israel's Ethnogenesis
Elman Service, Origins of the State and Civilization
Frank Frick, The Formation of the State in Ancient Israel
Peter Lemche, Ancient Israel
David Jamieson-Drake, Scribes and Schools in Iron Age Judah
Israel Finkelstein, The Quest for Historical Israel
William Dever, Who Were the Early Israelites...?
So--here's some of the evidence we look for when we are talking about the existence or non-existence of particular social systems of various levels of social complexity:
Furthermore, when we talk about an "early Israel", we are necessarily primarily going to be talking about the central highlands in Iron Age (i.e., 1200-586 BCE) Palestine.
The first bit of evidence we have for "Israel" is what's known as the Merneptah Stele (ca. 1204 BCE), commissioned by Pharaoh Merneptah (obvi.). This Egyptian Stele mentions "Israel" and marks the word with what we call a "determinative" (meaning a symbol that classifies the following word in a certain semantic word family). The determinative used with the name Israel is the one for a people group [as opposed to "nation"]). This epigraphic evidence is still very ambiguous and we're entirely certain what the determinative means--but we're certain that it mentions Israel directly.
Second, we have to understand that there was a massive social system collapse at the end of the Late Bronze Age (i.e., 13th century BCE). This caused a settlement shift from the Palestinian Shephelah (i.e., the lowland country to the west of the highlands) to the central highlands. At this point, we're probably talking about some kind of nearly egalitarian (true egalitarianism is pretty impossible to suggest in my opinion), kinship based society. This kinship system eventually led to the elevation of a particular family line where the pater familias of this line became a sort of paramount or "big man" who wound up running the system for a while. His family line then became sort of charged with what Service calls "mana"--essentially, leadership becomes somewhat hereditary at this point as the social system begins to stratify a bit more and social elites are starting to be a little more segregated from the regular, terrace farming and pastoral based economy.
I should point to "Yoffee's Rule"--if you can argue whether a particular social system is or isn't a state, then it isn't. I'm going to avoid trying to define state in a taxonomic fashion (because people like Service were rather obsessed with this silliness and it doesn't really get us anywhere because everybody makes the state look like what they want it to).
A few little notes about the biblical text--I'm convinced that Saul, David, and Solomon were historical individuals (i.e., they were not made up like a lot of people -- especially those over in /r/atheism -- would like to believe). Saul's enthronement was likely an instantiation of some kind of regularly cycling chiefdom where new chiefs were selected on a somewhat regular basis (see Jeremy Hutton's article comparing the Ahansali in Morocco as a sociological analogue to help reconstruct what's happening in 1 Samuel 10:17-27). We probably don't have a "state" until David comes along (historical evidence for a David: the Tel Dan Stele).
EDIT: I derped on the formatting