"A young man stands in his bedroom. It just so happens that today, the 13th of April, is this young man's birthday. Though it was thirteen years ago he was given life, it is only today he will be given a name!"
Good Morning, Crono! Happy birthday! Come downstairs and open your presents!
This trope is when the story more or less begins with the announcement that it's the main character's birthday. Often it isn't just any birthday, either; it's the day the hero finally reaches adulthood.
Why is this? Well... birthdays represent the start of a new chapter in one's life, which is suitably fitting for the beginning of the story itself. And it's a convenient way to establish how old a character is.
This can let the story start off with people deciding that it's time for the hero to learn an important secret about themselves. Or a mysterious character giving a bizarre, plot -related birthday present, amongst other things.
Tends to overlap with some of the other Birthday Tropes, especially Dangerous Sixteenth Birthday. Doubles as Easing Into the Adventure. Contrast with A Birthday, Not a Break, where we only learn it's the hero's birthday after the event. For the literal day of the hero's birth, see A Hero Is Born.
The Ruby/Sapphire saga of Pokémon Special begins on Ruby's 11th birthday, when he and his mother move to Hoenn and he runs away from home.
Also inverted: The same saga ends 80 days later, on Sapphire's 11th birthday.
Pokémon: Trainers are able to get their license and begin their journey on their tenth birthday.
In the Marvel Universe, most mutants gain their powers during puberty. In a surprising number of cases, this happens specifically on their fourteenth or fifteenth birthday.
Scion kicks off with Ethan's eighteenth birthday wherein he is thrust into combat against Bron.
The Empath: The Luckiest Smurf novel begins on the main character's 150th birthday, which turns out to be the day when he is finally released from being in Psychelia, and most of the Smurfs in the village are preparing for his birthday and his arrival home. 150 years for a Smurf (particularly a male Smurf) is basically the age of entering adulthood, as in The Smurfs cartoon show.
In "Days Of Future Smurfed", Empath celebrates his 1,150th birthday with his great-grandson, which ends up being the day that he dies, and his great-grandson becomes Traveler Smurf.
Sixteen Candles. Subverted in that everyone in her family forgot Samantha's birthday.
Lord of the Rings begins with (well, slightly before, but the Birthday-Party is the first major event in the story) Frodo's 33rd (and Bilbo's Eleventy-first) birthday. Since hobbits are long-lived, 33 is their coming-of-age year.
In the YA novel The Silver Crown, the heroine gets the titular crown for her birthday.
Harry Potter learned he was a wizard at more or less the hour he turned 11, and most of the other books started on or near his birthday.
Lirael opens on Lirael's fourteenth birthday. To her it's a humiliation because she has not yet had the Sight, or been Seen with the Sight, and is therefore still a child among the Clayr. However, the events of the day do change her life dramatically.
Skyward Sword downplays this, as it is the 25th anniversary of the Knights' Academy that Link is attending, but it is also the day he graduates to the Senior class. Much like the Sonic Generations example below, it is also the 25th anniversary of the Legend of Zelda series.