An archaic form of the name of the last day of the week in Middle-earth. Etymologically, it descended from the ElvishValanya, 'Day of the Valar', so the element Hih ('High') refers to the High Ones of Arda and their kind. Though its name is conventionally translated as 'Friday', as the last day of the week Hihdei was actually more akin to modern Sunday. It was considered the 'chief day' of the week, and its afternoon and evening were traditionally given over to leisure and festivities.
Note that Hihdei is the original form specified by Tolkien, but many editions of The Lord of the Rings spell this ancient name as 'Highdei'.
Notes
1
We do not have precise dating for archaic day names like Hihdei, except for a comment that they were in use 'at least nine hundred years' before the end of the Third Age (according to Appendix D to The Lord of the Rings). They were therefore in use in about III 2100, and at some point between that time and the end of the Third Age names like Hihdei evolved into the more recognisable Highday and its equivalents.
2
Lacking a direct explanation of how to pronounce the word Hihdei, our best approximation would be to follow Old English (the language from which the word originates). On that basis, both the h sounds would be pronounced as something close to Scots ch in the word loch, and the i sound would be short, as in 'him' rather than modern 'high'. The final -dei would be pronounced like the modern word 'die'.