Mary Shelley's Frankenstein is exists as both a Romantic and Gothic novel. Based upon its existence as a Romantic novel, the inclusion of elements of nature are of the utmost necessity. Therefore, the inclusion of a wet and dismal atmosphere remind readers that nature is always present.
As for the use of the wet and dismal weather's alignment to the Gothic, dreary weather supports the horror and mystery which surrounds the typical text of this movement. Given that lightening exists as the singular event which pushed Victor to pursue his study in science, the image needs to be continued throughout. At times, the lightening (accompanied by rain) illuminates the outline of the Creature. Through this, Victor is constantly reminded of his abomination and the threat the Creature possesses to him.
In the end, the dismal and wet weather add to the already horrific nature of Victor's experimentation. Compounding this, readers should come away from the novel in the same way Coleridge's wedding guest comes away from his conversation with the mariner--realizing the omnipresent power of nature.