By "proof," I mean that a claim has been established beyond a reasonable doubt, so that any reasonable person who examines the evidence will come to the conclusion that the claim is true.
What kind and quantity of evidence constitutes proof (in this sense) of a historical claim?
While I assume you are talking about big-H History in your question (the discipline working with the documentary record), I'll address how archaeological claims are evaluated since archaeologists are ultimately making (small-h) historical claims about the human past.
Much like with the documentary record, the archaeological record is partial and incomplete. This fragmentary nature of the record means that the standards of proof applied to other sciences like biology or physics, or even sociology, are unsuitably applied to archaeological evidence. The specter of small sample sizes and missing evidence forever haunt archaeological research.
As such, traditional statistical induction is often a poor match for archaeological data. When your samples are so small or heavily impacted by formation processes (those processes which govern what materials survive in the archaeological record), your ability to make traditional statistical inferences is limited, though I should stress not impossible. Furthermore, the type of historical claim archaeologists (and historians) are typically interested in are not generalized or universal statements about a specific type of data, the kind of statements usually provided by statistical inference. Rather, archaeologists are generally interested in synthesizing distinct lines of evidence into an explanation of past human behaviors.
Instead, archaeologists tend to stress that the best and most robust explanations of archaeological data are those which explain multiple lines of evidence. Alternatively, the more lines of evidence which support a particular conclusion, the more robust that conclusion. Fogelin (2007) argues that the type of inductive reasoning known as inference to the best explanation best characterizes this type of archaeological argumentation, based on multiple lines of evidence. This type of reasoning involves identifying the argument or interpretation which best explains the available data. That is to say, that the argument which explains the greatest quantity of evidence and most diversity of evidence is the best argument, and the best argument is the most likely to be true. In this case, "best" is of course a target that moves as new data is generated, but this is true of statistical induction as well.
As Fogelin highlights, this is the type of evidence most often employed in detective stories or in forensic cases. Sherlock Holmes reaches a conclusion about who committed a crime based on which culprit best fits all the available pieces of evidence, however disparate or unrelated they may seem. A similar line of argumentation is employed frequently in law. Likewise, archaeologists evaluate an argument by asking if the proposed behavior of past people fits the preponderance of the evidence.
This type of reasoning is important for two reasons. First, this type of argumentation helps us reach the types of synthetic arguments archaeologists (and historians) are typically interested in. Following an inference to the best explanation, unlike in statistical induction, disparate pieces of evidence are linked together into the overall interpretation and this actually strengthens the argument. In statistics, increasing the number of inferences actually weakens our certainty.
Second, because of the incomplete and partial nature of the archaeological record it is exceedingly difficult to "prove" that something didn't happen. This is a weakness that is frequently exploited by pseudo-archaeological claims, asserting that archaeologists cannot demonstrate that aliens didn't build the pyramids and therefore that their interpretation deserves an equal seat at the table. Archaeologists scoff at these claims precisely because pseudo-archaeologists are not evaluating the evidence using archaeological reasoning: while we cannot disprove an alien origin for the pyramids, that interpretation is far from the best explanation given all the available data. As such, the interpretation is rejected.
All this is to say that there isn't really a standard of proof in archaeology, in the sense that an argument can always receive additional support. There is no cutoff or threshold of evidence necessary for something to be considered true in archaeology. That isn't to say that there are not better and worse arguments, however. For instance, while two claims about different subjects may both be the best inference given the available data about that subject, if there is a considerably greater quantity and diversity of data about one of those subjects then that argument has met a higher standard of proof.
For instance, archaeologists nearly unanimously agree that the preponderance of the evidence indicates that the Americas were populated by migrations of people from eastern Siberia. Linguistic, genetic, geological, and a variety of archaeological evidence all support this assertion. In contrast, the exact route by which this migration may have happened is still debated. The current preponderance of evidence suggests that a coastal migration along the west coast of North America is the most likely route, but the quantity and diversity of evidence in support of any particular route out of eastern Siberia is much less than the quantity and diversity of evidence suggesting eastern Siberia as an origin for that migration. As such, while both the eastern Siberian origin and coastal migration interpretations are their respectively best explanation, one is significantly stronger than the other.
Edits: Typos and formatting.
Sources:
Fogelin, Lars. 2007. Inference to the Best Explanation: A Common and Effective Form of Archaeological Reasoning. American Antiquity, 72(4):603-625.
There is no golden rule of ‘proof’ accepted by all historians that can be used to sort fact from fiction. For one, different historical fields deal with different types and quantities of evidence, so there is no single approach that is going to work across contexts. For another, a lot of what historians are trying to do is build narratives and explanations from the limited evidence available to them. This inevitably requires interpretation, and is therefore always going to be subjective to some extent. As a result, historians disagree on big and little things about the past – this is entirely normal and inevitable.
That said, I can walk through the kind of thought processes I go through when trying to evaluate specific claims.
Do we have primary sources that are unrelated to each other which describe the same thing? If we have multiple independent sources from different perspectives that confirm the same thing, that is a basis for establishing what happened. For instance, if we want to know about what happened in a battle in the past, do we have sources originating from both sides that confirm what happened? If these sources agree on a version of events, then that version of events is more likely to be accurate. Even better might be a source from a third party that wasn’t directly involved. The more independent sources that attest to something, the more confident historians are that it happened. This is also vital as it’s rare that any single source will be able to cover everything – they might have only seen part of the battle, or been busy at a crucial moment, or perhaps something appeared different from where they were. By having a series of independent sources from diverse perspectives, it becomes possible to stitch together a fuller picture of what actually happened. This, however, involved making judgements about which bits to trust before others, which leads us to…
What ways might we expect any given source to be biased? What sort of institutional/personal perspective do they have? How might this affect what has been reported (or left out)? The goal isn’t to find some mythic, unbiased source, but rather use our knowledge of each source’s biases to inform how we treat it. For instance, in the hypothetical battle above, each side might have an interest in portraying the other in a negative way, or an author could be seeking to blame someone on their own side for a broader failure. Maybe the winner wants to exaggerate the numbers of the defeated enemy, to make their achievement look better. Even a neutral third party has their own agenda that shapes the way they report events. Understanding these biases allows historians to weigh different versions of events, and come to a conclusion when independent sources don’t agree on something.
Part of the advantage a specialist historian has is that we tend to know a lot about our subject matter and its immediate context - or, put another way given that we're talking about knowledge production, we are aware of a great deal of other evidence apart from what we're using at any given moment. That means that we are well placed to consider whether a particular claim fits into a broader picture. Part of this is judging plausibility – if we know that a particular weapon hadn’t been invented yet, we might know to be suspicious of an account that claims it was used in our battle. More broadly, does it contradict other things we know about the time and place? The hypothetical battle might have been part of a broader war, and therefore fits a pattern – in other words, a battle having taken place is fundamentally unsurprising. Or the opposite – it might have been an era of peace, and a battle would be unexpected. If it doesn’t fit the wider context, that doesn’t mean we should assume it didn’t happen – maybe our original picture was wrong, and needs to be revised – but perhaps we need to do a little more digging, and think about how this evidence can be best reconciled with other evidence.
It should be noted that I’m coming at this from the perspective of a modern historian – my problem is usually too much source material, not too little. Even for relatively recent history, however, most historians would deal with subjects about which there is uncertainty, although I suppose that reflects the practice of history itself – we get paid (well, sometimes) to find stuff out, not to rehash what we already know.