Eurythmics is one of the few bands named after a teaching method, and the method is older than the band by most of a century.
The method is eurhythmics, created by Émile Jaques-Dalcroze, a Swiss composer and music teacher born in 1865. Encyclopedia.com describes it as a system of training that linked musical rhythm with movements of the body. He taught it in Dresden before the First World War and founded his own institute in Geneva in 1915. The word itself is Greek-derived. Encyclopedia.com says eurhythmy comes from a Greek word meaning "good proportion or rhythm," and that it had earlier been used for architectural harmony, graceful posture and a regular pulse. Etymonline dates the adjective eurythmic to 1831, from Greek eurythmia, "rhythmical order," built from eu-, "good," and rhythmos, "proportion, symmetry."
The band came much later. Annie Lennox and Dave Stewart left the group The Tourists in 1980 and needed a name for what came next. Wikipedia says they named themselves after the teaching exercise system Lennox had encountered as a child. American Songwriter describes it as a learning exercise in which music was taught through movement and dance. Their debut album, In the Garden, followed in 1981, and Sweet Dreams (Are Made of This) in 1983.
Two details are shaky. The first is the history of the method. American Songwriter calls it a late 19th-century exercise, while Encyclopedia.com places its development in the early 20th century. The sources do not agree, so this page does not give a year for its invention. The second is who proposed the band's name. American Songwriter says nothing about who came up with it, and we found no source that does.
Both spellings are real. The method is often written eurhythmics, with an h, and Etymonline lists eurythmics, without it, as the variant for the same system. The sources do not say whether the band chose its spelling on purpose.