Prion diseases arise spontaneously fairly often. There are only ~170 total cases of human infection by mad cow disease, but there are ~50 cases/year of Creutzfeldt-Jakob disease, which is a spontaneous strain of prion diseases in humans. There is a strong case that Alzheimer's and Parkinson's diseases (among others) are similar in nature, in which case it's a much more frequent occurrence.
However, they are the consequence of a misfolding and aggregation of a naturally occurring protein produced in our brains. In this sense, they are nothing like the complexity required to produce a de novo virus. The closest viral analogy would be retroviruses, which could insert their genome into your DNA, die off, and then possibly spontaneously revive at a later date.