I was wondering, I heard someone claim that the word 'day' is mentioned 365 times in the Quran and that this constitutes a numerical miracle. Is it true?

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger. 

The word ‘day’ is mentioned in the Quran with the definite article (al) and without it and as part of an adverbial phrase more than 400 times. You may refer to some Quranic dictionaries such as Al-Mu’jam Al-Mufahras and others.

Allah knows best.