What does the H as in h-hour stand for?

It just means "hour." In the military, a lot of things are preplanned--say, the president is coming to our base to award Sergeant Jones the Medal of Honor for saving his whole unit. We won't know the exact time he'll be there until two days before the event, so we draw up an operational plan that says things like "two hours before he arrives, we need to inspect the battalion dining facility to ensure the president and first lady's favorite foods are prepared and that they taste good," "six hours before he arrives, we need to move the generator to the parade field so there will be power for the ceremony" and "the day before he arrives, inspect all the 30-kW generators in the battalion. Pick the best one and make it run perfectly and look new." Soldiers hate to write, so this would actually be written as follows: H-2: Inspect DFAC.
H-6: Deploy GENSET to BN PT field
D-1: Select and prep 30K GENSET When the White House calls the unit and tells us he's going to perform the ceremony at 11am and he's going to spend half an hour visiting with the troops before he has lunch with them, we now know to check the chow hall at 0900 and take the generator to the PT field at 0500. As for the generator, we're going to pick it out really early in the morning and two soldiers are going to spend all day working on it.