1976.
It would've been much cooler to have come of age around the beginning of the 90s instead of around the turn of the century.
In many ways, it was a great year to be born. Graduating college into a boom was a big plus, and the 1990s were better than the 2000s, which feel like what I was told the 1970s were like. In my own case, if I'd been ten years older there's a decent chance I'd be dead, whereas if I were ten years younger I wouldn't have internalized the message of the AIDS crisis to the extent that it has protected me.
There were a few major drawbacks:
1. Not having the Internet until college.
2. It makes you 18 in 1994. Think of how that feels as a Democrat--paying for the sins of earlier generations and not being able to catch a break in election after election. It's one reason I empathize with our young conservatives here, who can't help coming of age at the end of a conservative cycle. (Yes, I know, anything can happen in 4 years! I doubt it, though.)
3. If you didn't buy a house early on, you got locked out of the boom.
For myself, I would have chosen 1981. You still graduate into a good economy, but you have better computer games and the Internet at an earlier stage. That would have made high school so much better.