I'm not totally sure the origin of celebrating someone's birthday, but as early as the ancient Roman Empire (~1st-2nd c. AD), it was a fairly widespread practice. According to tablets unearthed at the Vindolanda Fort archaelogical site on the northern British frontier, one in particular reads:

"Claudia Severa to her Lepidina greetings. On 11 September, sister, for the day of the celebration of my birthday, I give you a warm invitation to make sure that you come to us, to make the day more enjoyable for me by your arrival, if you are present (?). Give my greetings to your Cerialis. My Aelius and my little son send him (?) their greetings. (2nd hand) I shall expect you, sister. Farewell, sister, my dearest soul, as I hope to prosper, and hail. (Back, 1st hand) To Sulpicia Lepidina, wife of Cerialis, from Severa."

You can read more about this tablet in particular here. Herodotus also mentions the Persians as celebrating birthdays as well, and that was far earlier than the Romans!