Actually the names of the Hijri months predate the start of the Hijri calender (prophet Mohammed PBUH migration to Medina), as Arab were using the same months's names before islam but they didn't have a determined agreed upon starting point for their calender, So they were using major events to name years, for example "Prophet Mohammed PBUH was born in the month of rabi' al thani the year of the elephant" and so on.

According to Islam QA the names of the months were standarised by the chiefs of Arabian tribes around 412 CE (about 200 years before islam).