Questions & Answers about Ночью у меня начался кашель.
Ночью is the instrumental case of ночь (night), and here it is being used adverbially to mean at night / during the night.
This is a very common Russian pattern with words for times of day:
- утром = in the morning
- днём = in the daytime / during the day
- вечером = in the evening
- ночью = at night
So in this sentence, Ночью tells you when the cough began.