The Historical Method is based on three fundamental steps, each of which has its own techniques:
1. Heuristic - This is the identification of relevant material to use as sources of information. These can range from the obvious, such as a historian of the time's account of events he witnessed personally, to the much less obvious, like a medieval manor's account book detailing purchases for the estate. Everything from archaeological finds to coins to heraldry can be relevant here. The key word here is "relevant", and there is a high degree of skill in working out which sources of information are pertinent to the subject in question.
2. Criticism - This is the process of appraisal of the source material in the light of the question being answered or subject being examined. It involves such things as determining the level of "authenticity" of a source (Is it what is seems to be?), its "integrity" (Can its account be trusted? What are its biases?), its context (What genre is it? Is it responding or reacting to another source? Is it using literary tropes that need to be treated with scepticism?) Material evidence, such as archaeology, architecture, art , coins etc needs to be firmly put into context to be understood. Documentary sources also need careful contextualisation - the social conditions of their production, their polemical intent (if any), their reason for production (more important for a political speech than a birth certificate, for example) , their intended audience and the background and biases of their writer (if known) all have to be taken into account.
3. Synthesis and Exposition - This is the formal statement of the findings from steps 1 and 2, which each finding supported by reference to the relevant evidence.
This involves abstracting from the messiness of practice, but that's exactly what you have to do to look at methodology. He also considers ways in which people go wrong. I think the biggest reason people go wrong is that it's just plain hard study: in history you are dealing with so many things, of so many different kinds, that you really have to sit down and study not to go astray. But given that, the other reasons he gives end up being amplified. And because of the level of difficulty correcting serious errors and misrepresentations can be extremely labor-intensive -- which might account for the widely recognized fact that historians can get a little irritated. And, of course, it's worth pointing out that while O'Neill, an atheist himself, is considering atheist failings in the regard, most of the things he says generalize very, very easily to other groups.
One of the things that I think would be interesting to look at in some detail is the historian's need for what might be called, for lack of a better term, cautious charity, which is related to what O'Neill is talking about when talking about objectivity. The historian doing serious history can't often take things at face value because other evidence may change the lay of the land quite considerably. For instance a document that seems quite straightforward and whose claims are plausible could be shown to be subtly polemical in light of contextual evidence; or it could happen that further comparison with documents would show that there is a highly conventional, formulaic aspect to it that has to be taken into account. But at the same time the historian, possibly more than anyone else engaging in intellectual inquiry, has to keep in mind that perspectives are myriad, and while this doesn't imply that they are all right, it does require that we recognize how relevant features of our own perspective may be influencing our reasoning. For instance, slavery is a morally pernicious practice, but our recognition that it is so needs also, if we are going to study the history of slavery, the recognition that we find it so easy to accept this in part because we have lots and lots of energy-slurping machines that contribute to our lives more than slavery possibly could -- our rectitude costs us nothing. This is not going to be the case with others, and this fact at least has to be taken into account. People usually do things because it's what seems to make sense at the time, given their context, and if you're going to understand anything in history, you have to make an effort at least to see why they thought it made sense. So there's a real need for an attitude of cautious charity: recognize these historical periods you are studying as being full of real people, not caricature devils or idealized fantasies, and letting neither liking nor distaste interfere with taking the evidence into account.
This is particularly important when we are drawing moral conclusions. History is, indeed, something that can contribute to our moral reasoning; but it's useless as such unless we are careful to make sure it's the evidence and rational inference that's doing the work.
