What's the grammatical name of the end of the day here, and what is its grammatical function?
The sentence is this:
There was always a huge quantity of food left over at the end of the day.
There was always a huge quantity of food left over at [the end of the day].
The phrase the end of the day is a noun phrase. The Head noun in the noun phrase is the word end. It is functioning as the Complement of the preposition at. Some people would describe this as the end of the day being the object of the preposition at.
Just in case the question was actually intended to be "What is the name and function of at the end of the day?", that phrase is a preposition phrase. In terms of grammatical functions or grammatical relations it is a temporal Adjunct (read 'Adverbial').
At the end of the day is an adverbial phrase. The clue is the preposition ‘at’. Like any modifier, it modifies (tells us about) the event or action described by the verb. In this case, it tells us when it took place. So it is temporal.