A song dedication doesn’t mean it is written about or for a particular person. I just assumed it was about childhood to adulthood -- the realization that you are not as important as you might think you are.
Bono sang in Party Girl: "when I was three I thought the world revolved around me; I was wrong." Day Without Me: “I'm starting a landslide in my ego, look to the outside to the world I left behind: a day without ME". I capitalized: 'me' because the ego is all about 'the me' or 'the I', and the lyrics certainly point to the beginning of maturity, the start of the destruction ones self absorbed adolescent ego.
I read the comments from the web site listed, but I would just disagree here about the suicide thing, even though it's compelling. It's not Bonoish. Lastly, the whole theme of Boy is about the emergences of an adult realization of ones self. Brilliant concept, and only rock album like it. I think it’s simply about the death of a childlike ego.
From "Into the Heart: The Stories Behind Every U2 Songs", by Niall Stokes:
"As singles go, this was a song with a big theme - or series of themes - which were only sketchily executed. A guy Bono knew - "he was an acquaintance of a friend of mine, Sean d'Angelo" - had tried to commit suicide. "In fact I went up to the hospital with Sean to see this guy..."(Bono said)....
The suicide attempt played in Bono's imagination and emerged in 'A Day Without Me', with the protagonist looking back at the world he has "left behind" from the perspective of the grave - or more likely a vantage point somewhere above the graveyard, as he watches the funeral and takes note of those who haven't shown up. "I was fascinated by the thought: would it make any difference if you did commit suicide?" Bono recalls.
Typically, that is just an undercurrent. The song also touches on the theme of insanity, and on the caollapse of the self, reflecting another common teenage insecurtiy."