Obviously there's no objectively right or wrong answer here, of course -- it's all about personal views. I'm just curious about everyone else's thoughts.
Some will no doubt say "When Fender sells to CBS" so I thought I'd get that out of the way first.
For me, I have a few:
1) Once you put humbucker(s) on it, you've crossed the line into 80's Hybrid Territory. (Stack-buckers don't count because the whole point of stacks is to retain the sound and look, just lose the hum, and some stacks do this extremely well.)
2) If you put a Floyd Rose on it -- well...yeah, an 80's Hybrid thing (because a regular humbucker usually accompanies this mod) but...well, to me it's borderline. But some have put a Floyd on and left the single coil bridge in. But then again, a Floyd will make it sound different too. So this one's too close to call for me.
3) If you put a new neck on it and said new neck has an ebony fingerboard, you can no longer call it a Strat.
4) If said new neck's headstock shape is wildly different from the Fender shape (I know it can't be exact), the line has been crossed.
Hardly earthshaking stuff, just small talk to jump-start me this groggy morning...