Herion was a descendant of the House of Húrin, the hereditary line of the Stewards of Gondor. At the time of Herion's birth, his grandfather Mardil was Steward to KingEärnur of Gondor, and because of his bloodline, Herion would expect to inherit the same position in time. In the year III 2050, however, when Herion was just thirteen years old,3 the King was tempted by the Witch-king to answer a challenge in Minas Morgul. Eärnur entered the dark city and was never seen again, and so the line of the Kings of Gondor came to an end. Thus Herion's grandfather Mardil Voronwë became the first of the Ruling Stewards of Gondor in the King's place.
The position of the Steward as the first servant to the King had become hereditary long beforehand, so it was natural that Mardil would in time by succeeded as Ruling Steward by his son (and Herion's father) Eradan. During Eradan's time as Steward, the Nazgûl who had destroyed the Kingship ceased to offer a direct threat to Gondor, because their secret master, the Necromancer of Dol Guldur, had fled for a time into the East.
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.
2
'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 it would be some years before Herion's grandfather Mardil would establish the line of the Ruling Stewards. 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.
3
Herion is notable as being the last of the Ruling Stewards who could remember a time when a King ruled over Gondor. His own son Belegorn was born (according to our best records) in III 2074, more then twenty years after the loss of Eärnur.