From New Advent Encyclopedia:
So also in the Hieronymian Martyrologium (c. 590), besides the constantly recurring natale applied to the festivals of martyrs we have, e.g. on 2 August, In antiochia natalis reliquiarum Stephani protomartyris et diac. None the less a certain stress was often laid in Christiansermons and in mortuary inscriptions upon the idea that the day of a man's death was his birthday to a new life. Thus St. Ambrose (Serm. 57, de Depos. St. Eusebii) declares that "the day of our burial is calledour birthday (natalis), because, being set free from the prison of our crimes, we are born to the liberty of the Saviour", and he goes on "wherefore this day is observed as a great celebration, for it is in truth a festival of the highest order to be dead to our vices and to live to righteousness alone." And we find such inscriptions as the following
PARENTE FILIO MERCURIO FECE
RUNT QUI VIXIT ANN V ET MENSES VIII
NATUS IN PACE ID FEBR
Where "natus in pace" clearly refers to eternal rest. So again Origen had evidently some similar thought before him when he insists that "of all the holy people in the Scriptures, no one is recorded to have kept a feast or held a great banquet on his birthday. It is only sinners (like Pharaoh and Herod) who make great rejoicings over the day on which they were born into this world below" (Origen, in Levit., Hom. VIII, in Migne P.G., XII, 495). Naturally a certain amount of confusion resulted from this use of the same word natalis sometimes to signify natural birth, sometimes the passage to a better life. The former was consequently often distinguished as natale genuinum, natale de nativitate, the latter as natale passionis or de passione, sometimes abbreviated as N.P.