'A stroke of .....' is a set phrase that goes with a very limited number of nouns, of which 'luck' and 'genius' are the most common. 'Stroke' a noun that comes from the verb 'to strike'. 'A stroke of ...' suggests an image similar to a flash of lightning striking a tree, for example - sudden and unexpected. A stroke of luck is when something lucky suddenly and unexpectedly occurs. A stroke of genius is maybe when a brilliant idea hits you.

"stroke of" as it is used in English sentences, generally refers to a movement of force or the action of some force. For example, one common usage combines the phrase with LUCK, as follows. He only won the race because of a stroke of luck, not because he was a more skillful competitor. Other examples: His invention was demonstrative of a stroke of sheer genius. I created that painting as the consequence of a stroke of artistic inspiritation.