What is the derivation of the names of the Islamic months, and what is the ruling on adding them to the calendar for people to learn from, especially since these months existed before Islam?

The Arabs used names for the lunar months before Islam. These names changed and were unified to take their current form since the late fifth century CE. Muharram was named to emphasize its sacredness (tahrim); Safar was named for the emptying of houses as people left for fighting and travel; Rabi' al-Awwal was named for their settling down in it; Jumada was named for the freezing of water therein; Rajab was named for reverence (ta'zim); Sha'ban was named for the dispersion (tasha'ub) of tribes; Ramadan was named for the intense heat; Shawwal was named for the camels raising (shawl) their tails; Dhul-Qa'dah was named for their refraining (qu'ud) from fighting therein; and Dhul-Hijjah was named for their performing Hajj therein. There is no harm in mentioning these meanings and their etymological origins for educational purposes.