As the direct descendant of Isildur himself, it could be said that Arvedui had the stronger claim, but Pelendur moved the Council to elect Eärnil, who took the Kingship as Eärnil II. All available evidence suggests that Pelendur continued to serve as Steward under the new King.
Pelendur was a highly influential Steward, who accrued considerable power to his office. As well as effectively acting as regent during the interregnum, it was during his time that the office of Steward became hereditary. When Pelendur died, he was succeeded as Steward by his son Vorondil, and his grandson Mardil was to become the first of the Stewards to rule Gondor in his own right.
The date of Pelendur's birth appears only in The History of Middle-earth volume 12, The Peoples of Middle-earth. It cannot therefore be considered completely reliable.
The -(n)dur ending of Pelendur's name clearly means 'servant' (as in his official Elvish title Arandur, meaning 'King's servant'). The initial pele(n)- element is less certain, but probably derives from the Elvish for a surrounded or fortified area, as in the Pelennor ('fenced land') that surrounded Minas Tirith. Indeed, Pelendur's name may imply that he was the 'servant of the (people of) the Pelennor'.