One of the advocates of describing evolutionary change as being "Non-Darwinian" is Lianne Gabora. However, check out what she means by "Darwinian":
To make writings on these matters less awkward, the terms ‘Darwinian’ and ‘selectionist’ are used as a shorthand for ‘by means of natural selection or a process that is algorithmically equivalent to it’. It would be wrong to interpret this as implying that Darwin never gave thought to evolution by means other than natural selection. He was, of course, immersed in the views of his day, and considered several possible explanations for adaptive change (e.g., his ultimately unsuccessful theory of gemmules). Moreover, although Darwin did not use the term neutral evolution, he acknowledged that evolutionary change can involve fixation of variants that confer no selective advantage over previous adaptations; nevertheless, neutral evolution is commonly referred to as non-Darwinian [70,75,123]. Similarly, although Darwin was not committed to the idea that all life evolved from a single common ancestor, processes such as horizontal gene transfer (that is, genes transmitted between organisms in a manner other than through traditional reproduction) are commonly referred to as non-Darwinian .With this definition of Darwinism, we are all non-Darwinians - including Darwin himself. Larry Moran is another advocate of this definition of "Darwinism". I am pretty sure that this is not a good way to define the term "Darwinism". This is straw man Darwinism. Only critics define Darwinism in this way: no advocates do so. Why is this bad? Critics have a weaker claim on defining terminology in a field than advocates do. They should mostly accept the terminology of advocates when criticising. If critics get to make up too much terminology, we get a polluted namespace, full of too many daft and useless terms.
Another complaint about "Darwinism" is that the term is used pejoratively by creationists:
Evolutionary biology owes much to Charles Darwin, whose discussions of common descent and natural selection provide the foundations of the discipline. But evolutionary biology has expanded well beyond its foundations to encompass many theories and concepts unknown in the 19th century. The term “Darwinism” is, therefore, ambiguous and misleading. Compounding the problem of “Darwinism” is the hijacking of the term by creationists to portray evolution as a dangerous ideology—an “ism”—that has no place in the science classroom. When scientists and teachers use “Darwinism” as synonymous with evolutionary biology, it reinforces such a misleading portrayal and hinders efforts to present the scientific standing of evolution accurately. Accordingly, the term “Darwinism” should be abandoned as a synonym for evolutionary biology.I don't see why scientists should adopt terminology based on a bunch of idiocy about theology and god. I'm regularly told that I can't use the term "intelligent design" - because that term has already been taken - by a bunch of idiots. As you can probably imagine, I am not impressed by this: idiots don't get to decide what terminology scientists use.
"Darwinism" might not be an ideal term. However, it seems to me that it has fairly comprehensively trounced its competitors in the struggle for survival of words. In terms of volume, even neo-Darwinism fails to compete. The "don't call it Darwinism" paper fails to come up with a decent alternative. The nearest thing it manages to offer is "evolutionary biology". This seems to fail to me. "evolutionary biology" is an application domain. Darwinism is a refutable theory. Those things are not really in the same category. Also, "evolutionary biology" is a topic confined to biology. When I use the terms "Darwinism" and "Darwinian evolution" I am talking about something that also covers physics and chemistry - and is emphatically not confined to biology.
"Darwinism" has been a pretty spectacularly successful meme. I think we should make the most of it.