A fellow atheist asked me some of these questions about metaphysics and I thought I'd answer them here for good measure.
• What exactly is metaphysics?
As I come to understand it, metaphysics is a branch of philosophy concerned with fundamental ontological claims that we do not have empirical evidence for. Once we have empirical evidence for it, it becomes physics. Until then, it's technically metaphysics. There are ideas for example that theoretical physicists come up with that make claims about reality that we do not have empirical evidence for, such as String Theory. They would technically be metaphysical, although many of them are in principle verifiable. I consider metaphysics to cover ontological claims that are both capable of being verified in principle, but yet haven't, and ones that aren't in principle verifiable.
• What constitutes a metaphysical claim?
It must be about the fundamental ontological nature of reality, but something we don't have empirical evidence for. So the claim "angels exist" is a metaphysical claim, and the claim, "the multiverse exists" is also a metaphysical claim. The difference between them is that one is derived from empirical evidence, and one is not.
• What is the methodology for assessing the validity of a metaphysical claim?
By applying logic and rational thought using the best available scientific theories and evidence.
• In what way is the term 'metaphysics' useful such that commonly accepted, unambiguously defined scientific terminology is not?
Metaphysics is a bit more like an umbrella term that can be used interchangeably with some scientific terminology, like a hypothesis, if it's about fundamental ontology. But metaphysics also covers claims that are far outside the domain of science, like claims about the supernatural. It seems that some atheists want to delegate the term metaphysics to only cover claims about the supernatural/paranormal and not have it overlap in any way with science. But to get rid of metaphysics because there are many different ways to define it, one would also have to get rid of the term "religion" because there are many ways to define it, and no consensus appears in sight. Metaphysics is not supposed to replace scientific terminology, its supposed to coincide with it when their domains overlap.