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!