A lightning storm was a meteorological phenomenon characterized by a high amount of electrical activity.
In 2233 of the alternate reality, a comm officer serving on the bridge of the USS Kelvin compared spatial effects surrounding the appearance of the Narada to a lightning storm, a description which was later noted in a dissertation authored by Christopher Pike. After hearing Pavel Chekov describe a 2258 anomaly as a "lightning storm in space", James T. Kirk immediately recognized the term, certain that it was the same phenomenon that had resulted in the death of his father. (Star Trek)
In 2364, when Wesley Crusher was testing himself in preparation for his psych test for the Starfleet Academy entrance requirements, he asked Worf's opinion as to what fears he could face. "Lightning storms" was one of the ideas Wesley thought of. (TNG: "Coming of Age")