What is the lightning coming out from Boruto's eye at the end of 13th episode of Boruto? Is this some power used by him unknowingly?
It is Dojutsu, the ability to detect things such as chakra which is generally unseen by a normal eye. At the age of 8, which is the age he is at currently within the anime, he is able to use it subconsciously, i.e. unknowingly but at the right moments. But if you watch the first episode where we see a teenage Boruto we see that Boruto is now able to use the Dojutsu, i.e. his eye on his own will.

The origins of this power are still unknown, and I can only speculate that this is somehow related to the seal that Momoshiki presents on Boruto's palm after Boruto defeats him.