The reason why the Hijri months are called by their well-known names
Question

How were the names of the Islamic months derived? Because I intend to add them to the calendar so that people can learn them, but I want to know the ruling on doing this, because these months, as far as I know, were known before Islam appeared.
Praise be to Allah.Before Islam, the Arabs used different names for the lunar months that were known at that time, until those names changed and became standardised throughout Arabia, and took their final forms at the end of the fifth century CE, during the era of Kilaab, who was the great-great-great-great-great-grandfather of the Prophet (blessings and peace of Allah be upon him).
The naming of these lunar months with the names that are known today is based on certain reasons and meanings from which the names of the months were derived, as was mentioned by the scholars. Al-Haafiz Ibn Katheer (may Allah have mercy on him) said:
Shaykh ‘Alam ad-Deen as-Sakhaawi stated in a book that he compiled, entitled al-Mashhoor fi Asma’ al-Ayyaam wa’sh-Shuhoor, that Muharram is so called because it is a sacred (muharram) month. I think that it was so called to affirm its sanctity, because the Arabs used to change it, making it not sacred one year and making it sacred another year. He said: The plural form of Muharram is Muharramaat or Mahaarim or Mahaareem.
Safar is so called because their houses would be devoid of them when they went out to fight or to travel, and it would be said “safira al-makaan (the place has become empty)” when people left. The plural form of Safar is Asfaar.
Rabee‘ al-Awwal is so called because they did not travel during that month (the word irtibaa‘, derived from the same root, means “not travelling”). The plural form is Arbi‘aa’, like the word naseeb (pl. ansibaa’), or Arbu‘ah like the word ragheef (pl. arghufah). The same may be said concerning Rabee‘ al-Aakhir as was said concerning Rabee‘ al-Awwal.
Jumaada is so called because water froze during that month (jumood means freezing).
And he said: In their calculations, the months did not rotate through the seasons, although there is some doubt concerning that, because their months were connected to lunar cycle, so they would inevitably rotate through the seasons. Perhaps they called Jumaada by this name the first time, when water froze because of the cold. The plural form of Jumaada is Jumaadiyyaat, like the word hubaara (pl. hubaariyyaat). The word may be either masculine or feminine, so one may say Jumaada al-Oola or al-Awwal, and Jumaada al-Aakhir or al-Aakhirah.
The name of Rajab comes from the word tarjeeb, which means veneration. The plural form is Arjaab or Rijaab or Rajabaat.
The name of Sha‘baan comes from the word tasha‘‘ub (scattering), as the tribes would scatter and disperse to raid others. The plural form is Sha‘aabeen or Sha‘baanaat.
The name of Ramadan comes from the word ramda’, which refers to intense heat. Another word from the same root is used to refer to the thirst of camels when they become thirsty. The plural form is Ramadaanaat or Ramaadeen or Armidah.
The name of Shawwaal comes from a word referring to camels raising their tails for the purpose of copulation. The plural form is Shawaawil or Shawaaweel or Shawwaalaat.
The month of Dhu’l-Qa‘dah – or, in my view, Dhu’l-Qi‘dah – is derived from the word qu‘ood (rest), as during this month they would rest from fighting and travelling. The plural form is Dhawaat al-Qa‘dah.
Dhu’l-Hijjah – or, in my view, Dhu’l-Hajjah – is so called because they performed Hajj during this month. The plural form is Dhawaat al-Hijjah.
End quote from Tafseer Ibn Katheer (4/128-129)
See also: al-Mufassal fi Tareekh al-‘Arab qabl al-Islam by the historian Jawaad ‘Ali (16/91ff)
There is nothing wrong with mentioning the names of the months, explaining their meanings and the sources from which they were derived, and the reasons why they are so called, as explained by the historians and linguists, especially if there is an educational purpose behind that, even though the origin may have been forgotten and no longer has anything to do with the names of the months, because they rotate through the seasons, as is well known, and therefore the meaning has nothing to do with the well-known shar‘i rulings on the months.
For more information, please see the answer to question no.  170242
And Allah knows best.