We have been on
DCL cruises delayed by fog out of Galveston twice, and both times there was compensation that I thought was adequate. Both times we were given a credit that was intended to cover the lunch that was missed on embarkation day--it was more than enough for our family to go out to lunch in Galveston. Then we also got some combination of the following: compensation for a missed port (had to skip Key West in order to get to
Castaway Cay on time); additional credit for general unhappiness I guess; very nice discount offered for future cruise (that could be combined with OBB for a REALLY nice discount).
Note that the best compensation was on the cruise where boarding did not even start until about 4pm and we missed a port. The one where it was just a couple of hours delay was less. But both times I felt it was adequate for the situation.
On one cruise we were substantially delayed getting off due to fog. They let us stay in our staterooms, showed movies, had some activities, and opened for lunch in Cabanas.