Disclaimer: I am not J.K. Rowling. I do not own Harry Potter.
Note: Just a brief idea which occurred to me but which went nowhere. Maybe it'll inspire someone else. Assumed AU.
Harry Potter was born in the early hours of the morning of July the 31st, 1980, whereas Neville Longbottom was born late in the evening of July the 30th of that same year. All but the most inbred of adult witches or wizards presented with those facts would deduce that Neville Longbottom must therefore have been born before Harry – a particularly important point when something such as a matter of prophecy is involved.
Except, of course, the facts are not always what they seem – especially not when something such as the speed of midnight is brought into consideration.
Harry Potter, you see, was born in Britain; Neville Longbottom was also born on British soil – but it happened to be British soil located in the Pacific Ocean, somewhere between Peru and New Zealand, where Neville's auror parents had been sent a week earlier on an exceptionally urgent and secret mission that required their personal presence. Neville Longbottom was born on Pitcairn Island, several hours after Harry, but what was still (locally) the calendar day before.
And where prophecies are involved, such twists of fate count for a considerable difference.
Author Notes:
And that's where this idea unfortunately ran out. I had a vague idea that only Albus Dumbledore would know that Neville was born after Harry, and that that would prove important with everyone else who knew about a prophecy (Voldemort included) thinking Harry was the 'born as the seventh month dies' target, but Albus believing it was Neville.
Anyway, maybe someone else will be able to take this idea and run with it.