What are some tips you have for doing well in a humanities interview at Oxford or Cambridge?

It's really useful to treat interview questions in the same way you would go about answering an essay question.
Let's take the example of if you were asked "what are some of the major drivers of inequality today?" Just like in the introduction to an essay, a great way to start would be to define the question's key terms, in this case inequality, and split it up into its different subcategories (racial, gender-based, economic, regional inequality etc). When asked such a broad question, you might then want to narrow the focus somehow. For example, you could acknowledge all of these different types of inequality, and the various processes which perpetuate them, but then choose to zero in on one particular aspect, making the argument that this is one extremely important form of inequality. It might also be a good idea to specify where your answer is focusing. Inequality's drivers may be very different in the US and in Thailand, for example. Interviewers are looking for this kind of specificity, an awareness of the nuance of these huge questions, and an answer which makes itself sharp and effective by not attempting to make broad generalisations which are easily contestable.You might also want to turn this into a challenge to the terms of the question. In this example, that might come in the form of arguing that there is no single "inequality", and so such a question cannot be answered without an acknowledgement of the various ways that inequality manifests itself. This shows an active engagement with the question and the sort of sharp analytical thinking they are looking for. Finally, any argument (in an interview or an essay) needs evidence to back it up. Although in an interview this is not essential, throwing in some key theorists or examples from current affairs, especially if these go beyond the standard A-Level syllabus, is a great way to show (without telling) wider engagement in the subject and an ability to apply knowledge to strengthen an argument.
KEY POINTS: Define key terms, acknowledge the breadth of the question and then state chosen (more specific) angle, challenge the terms of the debate and back up your argument with evidence.