COVID-19 patients can exhibit a wide range of clinical manifestations affecting various organs and systems. Neurological symptoms have been reported in COVID-19 patients, both during the acute phase of the illness and in cases of long-term COVID. Moderate symptoms include ageusia, anosmia, altered mental status, and cognitive impairment, and in more severe cases can manifest as ischemic cerebrovascular disease and encephalitis. In this narrative review, we delve into the reported neurological symptoms associated with COVID-19, as well as the underlying mechanisms contributing to them. These mechanisms include direct damage to neurons, inflammation, oxidative stress, and protein misfolding. We further investigate the potential of small molecules from natural products to offer neuroprotection in models of neurodegenerative diseases. Through our analysis, we discovered that flavonoids, alkaloids, terpenoids, and other natural compounds exhibit neuroprotective effects by modulating signaling pathways known to be impacted by COVID-19. Some of these compounds also directly target SARS-CoV-2 viral replication. Therefore, molecules of natural origin show promise as potential agents to prevent or mitigate nervous system damage in COVID-19 patients. Further research and the evaluation of different stages of the disease are warranted to explore their potential benefits.