In common with all the names from the Shire Calendar, Rethe comes from Old English. It derives from Hréðmónað, an Anglo-Saxon name meaning 'month of Hreða' (Hréða was a goddess whose name means 'Glory'1). This name is merely a 'translation' into archaic English of the 'true' name of the month that was used by the Hobbits themselves, but that original name is nowhere recorded.
Hréða's association with the month of the Spring Equinox, as well as her name 'Glory', presumably means that she was associated with the Sun in some way. The details of her history, though, are now almost entirely lost.