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, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
We do not know of any evidence indicating the merit and virtue of dying in the month of Muharram. There is no doubt that it is a blessed month and that it is recommended for Muslims to fast more frequently in this blessed month. Abu Hurayrah, may Allah be pleased with him, reported that the Prophet, sallallahu ‘alayhi wa sallam, said, "The most virtuous fasting after the month of Ramadan is the month of Allah, Al-Muharram." [Muslim]
It is not unlikely that Allah, The Exalted, would generously reward the person who dies during blessed times. Al-Bukhari allocated a chapter in his book Saheeh Al-Bukhari entitled "Chapter: Dying on Monday." Al-‘Ayni wrote, "This chapter underlines the excellence and virtues of dying on Mondays." [‘Umdat Al-Qaari] Al-Bukhari cited the following hadeeth therein: "‘Aa'ishah, may Allah be pleased with her, went to Abu Bakr, may Allah be pleased with him, (during his fatal illness) and he asked her, 'On which day did the Prophet, sallallahu ‘alayhi wa sallam, die?' She replied, 'Monday.' He asked, 'What day is today?' She replied, 'Monday.' He said, 'I hope I shall die sometime between now and the night.'" [Al-Bukhari]
Al-‘Ayni said, "Since the Prophet passed away on a Monday, then whoever passes away on a Monday, hopefully they shall be granted good (rewards) because they passed away on the day on which the Prophet died. Therefore, Monday has an advantage over other days in this regard."
Allah knows best.