While I understand your point, I find that a surprising number of people have issues with 12am vs. 12pm, often confusing the two. These same people tend to think midnight occurs at the end of the day.

Everybody who isn't American or British has issues with am/pm. It is so counter intuitive it is not even funny.

jorangreef is probably right, at least assuming that we get to colonise other worlds. I mean, what does it mean to talk about 'day' when you're on the ISS? How are Martians going to deal with the fact that they have a Sidereal Rotation Period of 24h37mins? Or the Moon, with it's roughly 700hr SRP.

Right on. But maybe a better time would be TAI - UTC without the leap adjustments.

Edit: We basically just need a logical clock though we may face problems reconciling this logical clock in a distributed space/time system.

Did you mean 0:00 is night and 12:00 is day?