Buffy: The Vampire Slayer - Who"s the Next Slayer?
Throughout season 7 Buffy gives many speeches to the potential slayers stating that if she dies one of them will be called upon to be the next slayer. But when Buffy died at the end of season 5, there was no slayer called upon because Faith is the active slayer, not Buffy. Which means if Buffy died, no slayer would be called upon.
This was explained in a "behind the scenes" book I bought.
At the end of the first season when Buffy dies for a few minutes in "Prophecy Girl" we find out in the second season that Kendra was made the slayer, technically making her the "active slayer".
When Kendra dies at the end of the second season we find out in the third season that Faith was called, making her the active slayer.
So when Buffy dies at the end of the fifth season, no new slayer is called because Faith, the "active slayer" is still alive and technically according to "the powers that be" Buffy has been dead since the first season.
When Buffy says this, it could either be because
1) She doesn't realize that after having already died, her death no longer calls on the next slayer
2) When she says "my death", she basically just means "a slayer's death".
Either way, it doesn't seem as though the characters (or at least Buffy) knows that once a slayer has died, she no longer carries that torch, but is still connected to the Slayer power.