The ‘birthday present’ is shorthand, used to describe a particular song that carries certain notoriety for an artist or group. This song has come to define the band/artist, regardless of its age, quality or potential embarrassment. When performing live, this particular track is virtually expected; failure to perform the number will lead to grave disappointment or even bitterness from the crowd.
The ‘birthday present’ obtains its name through its special gift-like quality that, if not received, tends to cause resentment.
If Led Zeppelin came to town they’d just *have* to perform ‘Stairway to Heaven’, it’s their birthday present!
The use of the word "blue" to refer to sexually explicit content was first recorded in Scotland in 1824. "Blue" meant "lewd" because prostitutes dressed in blue gowns.