I guess it depends on the context, what a person's normal routine is, the time of year and what time sunrise is.
Fot lots of hobby related meet ups, "first thing" would depend on what time sunrise is, and/or what time something opens.
If my partner was booking a tee time for golf "first thing" it would be at about 7.30am atm because that's when sunrise is. It could be 6am in the summer.
If I met my friend for a swim "first thing" before work, it would be at 6am when the pool opens.
If we were meeting for a walk on a day off, then I'd take "first thing" as anything from sunrise to 9am for one friend as she gets up early, and anything from 9am onwards (but definitely not before 9am) for a different friend who gets up later.
If we were meeting for shopping then "first thing" would be when they opened at 9am.
If I was booking an appointment at my beauty salon and asked them for an appointment "first thing", then they'd book me in for 8am because that's when they open.
OP isn't meeting her ex socially though, they're just arranging a time to meet up to drop their kids off. To me, times in the morning driving wise are:
5am onwards = first thing
6am - 7.30am = early morning
7.30 - 9.30am = rush hour (to be avoided like the plague if possible)
9.30am - 11am = mid morning
11am - 12pm = late morning
So many variables!