The name of the third month of the Shire Calendar, a name also used by the inhabitants of Bree. In the Shire Calendar, Rethe fell between Solmath and Astron, and was approximately equivalent to modern March (though in fact it ran from modern 21 February to 22 March). This same period was more commonly known among the people of Middle-earth by its Elvish names, usually as Súlimë (but as Gwaeron among the Dúnedain).
In common with all the names from the Shire Calendar, Rethe comes from Old English. It derives from Hrēðmōnað, an Anglo-Saxon name meaning 'month of Hrêða' (Hrêða was a goddess whose name means something like '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.
Notes
1
Hrêða'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.