Are there any organisms, extant or extinct, that have only one neuron?
In evolution things don't just pop into existence. So, it's unlikely that a single neuron originated by itself. The question is, what would that neuron do? It would communicate to at least one other cell, right? Otherwise the transmitted signal would be lost and useless. And where is the signal coming from? Neurons generate signals either by receiving signals from other neurons or, for specialized sensory receptors, through some changes in the environment. The receiving cell could be a muscle cell, so your minimal network would consist of a sensory neuron and a muscle cell. Upon a detected change in the environment, the neuron would signal the muscle to contract.

In evolutionary basic organisms we can find action potential like excitability and synapse-like cell-cell contacts in non-neuronal cell types. This hints towards a step-wise development of these functions before actual neurons arose (wiki, paper). Even one-cell-organisms have neuron-like features (sensory input dependent polarization, intracellular signal propagation, coordinated movement). If cell-cell communication or excitability are not limited to neurons, what makes a neuron a neuron? This is also a matter of definition, as discussed in the paper mentioned above.

What makes neurons unique is their property to propagate signals in a network. That's why the neuronal nets of jellyfish are usually seen as the first appearance of neurons in the evolutionary tree. While organisms without a nervous system are able to respond to their environment by signal transmission between non-neuronal cells, I am not aware of any organism that possess only a single cell which we would now call a neuron. At the stage where neurons arose, animals were already a more complex arrangement of specialized cells for different functions. You would therefore assume, that also the neuronal predecessors were a group of cells within the organism.