"EEG " means Electroencephalograph, which means recording (graph) of electrical signals (electro) from the brain (encephalo). The Liverpool physician Richard Caton discovered electrical signals by scratching around directly on the surface of animal brains in 1875. He also described variations in the activity when he flashed light at his animals.
The actual inventor of the electroencephalograph was the Austrian Hans Berger who in the early 1920s recorded human brain activity and amplified it. He also was the first to describe wavelike pattern, which he named alpha and beta waves. And, more significant, he also was first to describe unusual EEG-patterns in epileptic patients. Berger made hundreds of recordings to make sure he really saw brain activity, and he also eliminated the possibility that the waves he recorded originated in the skin or somewhere else (more). Berger committed suicide in 1941. He was twice considered for the Noble Prize, but Nazi Germany prevented it.