The oldest record in Scripture of someone who wanted to celebrate their birthday is that of an Egyptian Pharaoh at the time of the patriarch Joseph. One of the sons of Jacob, Joseph lived from around 1709 to 1599 B.C. and spent the majority of his life in Egypt. The account of this event is in Genesis 40.
Our "birthday" story begins with a baker and a butler who served Pharaoh. They were both prisoners for bringing the wrath of their ruler upon themselves. While languishing in prison they meet Joseph. A married woman had falsely thrown him into jail when her sexual advances were rejected.
One night, a few days before Pharaoh's birthday, both the baker and butler have strange dreams. In the butler's dream, he sees a vine that has three branches. He describes the dream to Joseph and states that he held in his hand the cup of Pharaoh. With cup in hand he then "took the grapes (from the vine) and squeezed them into the cup and gave it to him (Pharaoh)" (Genesis 40:11).
The baker then says to Joseph that he dreamt he had three baskets on top of his head. The top basket held Pharaoh's baked goods, where the birds were eating them (verses 16 - 17). What the dreams would ultimately mean for the butler and baker, as was foretold by Joseph under God's inspiration, would come to fruition three days later on Pharaoh's birthday. The butler was given back his job serving the ruler, while the baker was hanged (verses 20 - 22).
Some people have reasoned that since a hanging occurred on this day that it therefore is wrong to celebrate the day of a person's birth. This is a 'guilt by association' argument that does not make much logical sense. While one person lost his life when Pharaoh commemorated his birth, another one gained their freedom! Not only that, but it was eventually through the butler that Joseph's life was ultimately spared!
After HE was saved, Joseph goes on to save his entire family - the literal patriarchs of the twelve tribes of Israel - from famine in the land of Canaan (see Genesis 45 and 46)! Note also that NONE of this would have happened if it were not for God giving Joseph the ability to interpret dreams. Added up, what occurred because of a birthday would be a strong argument for keeping them since more good than bad happened on the day!
The only other mention in the Bible of a birthday is that of Herod Antipas (one of Herod the Great's sons). The account is in Matthew 14 and Mark 6. Briefly, Herod had thrown John the Baptist in prison because of comments condemning his marriage to Herodias. Although Herod wanted to put John to death his "fear of the people," who saw John as a prophet, kept him from doing so.
Herodias, who also wanted John killed, knew of her husband's attraction to her daughter, Salome. On the day Herod was to celebrate his birthday, she and Salome conspired to trick Herod so that he would HAVE to kill John. Salome's dance so pleased Herod that he promised her anything up to half his kingdom (Mark 6:23). She requested John's head on a platter. Though surprised at the request, Herod felt he had to fulfill it since he made his promise in front of all his guests. John, the voice in the wilderness the heralded the coming of Jesus, was then martyred.
The birthday was incidental to the overall desire of Herod and family to want to get rid of John. Using John's death on the day Herod decided to throw a party to celebrate when he was born as a reason to shun rejoicing over one's birth is another fallacious "guilt by association" argument.
In conclusion, the Bible does not say it is wrong for a person to celebrate a birthday - either their own of someone else's. There simply is no teaching in God's word concerning these events one way or the other. There is no scripture, from Genesis to Revelation, which states it is a sin to keep track of the passing years. Neither is there a single scripture that says it is forbidden for families to rejoice at a patriarchal father reaching a great age, or hugging and loving a child, giving them a gift and congratulating them on their special day! In the end, it is not a sin for a person to observe their birthday.