The date of Herion's birth appears only in The History of Middle-earth volume XII, The Peoples of Middle-earth. It cannot therefore be considered completely reliable.
'Son of the lord' seems to be the correct interpretation of Herion's name (from Elvishheru 'lord, master' and ion 'son') but it is hard to interpret why he would be given this name. At the time of his birth there was still a King in Gondor, and Herion's grandfather Mardil would not establish the line of the Ruling Stewards until Herion was thirteen years old. So, at his birth Herion was not the son of a lord, nor was there any real prospect of him becoming one. Perhaps he chose to rename himself after the line of Ruling Stewards began, possibly to establish his hereditary right to succeed to that office.