The philosophical neuron is the basic unit of the human nervous system. This is very similar to the definition of a neuron. But a philosophical neuron is needed because there are many philosophical questions about the relationship between the physical structure of our brain, and the phenomena of our mind. Many of these questions are about how a collection of discrete parts interact to create thought and feeling.
However, when asking a question such as "Why do the neurons in our brain experience consciousness, while the neurons in our knee do not?" there is the distraction and complication that the actual physical neurons, the long spindly things covered in myelin, are quite possibly not actually the basic functional units of either our brains or our minds. Because while a neuron can be seen as either on or off, the truth is that there are millions of processes going on inside the neuron. It is quite possible that the basic functional unit of the nervous system is one of the millions of proteins inside that neuron, or (as Roger Penrose has it) a microtubule storing information in the orientation of the water molecules inside it. Or perhaps the basic functional unit is larger -- an entire network of neurons.
All of which are technical theories of neurology, but are complete dodges of philosophical questions. So a philosophical neuron is a way to rephrase philosophical theories about the brain to be more general. Whatever we consider the basic physical unit of our nervous systems is a philosophical neuron, and can be interchanged with the term "neuron". It could be a standard neuron, a protein, a floating electrical potential, a network of actual neurons, or even (for Matrix-type scenarios) the robotic components of our computer brains that underlie our delusion that we are organic. In all of these cases, the physical structure is superfluous to the question, and therefore we treat the questions as being ones about philosophical neurons, rather than biological neurons.
The idea is very parallel to the idea of the philosophical atom. The atomic theory of classical times led to questions and paradoxes that caused debate for thousands of years. Over the last two hundred years, we have learned a lot about physical atoms, but the philosophical paradoxes of atomism remain. The answer to the paradoxes about the atoms indivisibility are not "But it is made of protons and neutrons!" The philosophical atom is still not made from smaller parts, no matter if physical atoms are.