Electric solar boats have not been around for a very long time in their current form, although there is an aspect to their history that predates the modern internal combustion engine. As with most things electric, the early origins of electric transportation were destroyed by the introduction of the gasoline engine.
These early beginnings date back to around 1890, when Europeans were experimenting with powering boats through electricity. This was a natural logic progression from the discovery of electricity and the knowledge that steam power could move boats forward. The earliest fully electric boat was the Victory, commissioned in 1905. It could carry over 300 people at one time, although the solar panels were by no means particularly efficient. It is debatable to what extent people realized the low efficiency of the solar panels on the Victory, but whatever the case may be it is indisputable fact that the Victory was barely seaworthy. In fact, it only ever went up and down the Thames River.
The modern history of the solar electric boat probably dates from around 1975 and once again takes place in Europe. The 1970s were a time of mass innovation in terms of transportation. This was due to the oil shocks which made the western world realize just how dependant on foreign oil they were. In the end, the solar electric boat project in the United Kingdom in the 1970s was cancelled because of the resurgence of cheap oil, but it did act as a baseline from which many of the modern attempts at solar boat construction started.