Birthdays are a recurring motif in 2001: A Space Odyssey; there's one birthday per chapter… sort of.
We get word of the first birthday when Dr. Floyd calls his daughter on the video phone:
SQUIRT: She's gone to the bathroom. Are you coming to my party tomorrow?
FLOYD: I'm sorry, sweetheart, but I can't.
SQUIRT: Why not?
FLOYD: Well, you know, Daddy's traveling. Very sorry about it, but I just can't. (2001)
Harry Chapin warned about this very thing, Floyd.
The second birthday is Dr. Poole's. While aboard Discovery One, Poole celebrates his birthday with a video message from his parents. They even bake him a good-looking cake, just to rub in how nasty that zero-G space food looks.
Interestingly, parents and child are always separated during their birthdays. Floyd can't make it to Squirt's party, and Poole's message is entirely one-way. He can't respond to the birthday well-wishes.
HAL's death provides us the third birthday. As Bowman removes his memory cards, HAL reverts back to who (or what) he was when he was first activated; he sings the song he learned that day. So… more of an unbirthday, really.
From the Cradle to the Grave
But didn't we say there was one birthday per section? We said "sort of."
There aren't any birthdays in the first and final sections, but there are two births. In the Dawn of Man section, the early hominids learn to manipulate tools, and in a sense, this is the birthday of the human race. When Bowman becomes the Star Child, we have another birthday. This time the scene depicts the birth of a species hereto unknown in the universe.
Note that these birthdays tie into the separation of parent and child we noticed earlier. Just like Floyd is separated from his daughter, these evolutionary births separate the individuals from their parent species.
The film's always reminding us of the cycle of death and birth/rebirth. For humans to exist, our early hominid ancestors have to become extinct. Or, in the case of that one hominid, viciously beaten to death. For the Star Child to come into existence, Bowman has to die. In HAL's attempt to succeed in his evolutionary path, he tried to kill his human creators but failed.
It's the circle—the circle of life.