The date of Narmacil's birth appears only in The History of Middle-earth volume XII, The Peoples of Middle-earth. It cannot therefore be considered completely reliable.
The root-word nar literally means 'fire' or 'flame', but it is often used figuratively to refer to the sun, or to the colour red. So, 'sun sword' or 'red sword' would be reasonable alternative translations. In any event, it is likely that Narmacil II ultimately took his name from his thirteenth-century ancestor, Narmacil I.