Why is weather an element of Gothic literature?

First, let's clarify what "the gothic" means in literature of this era. A good definition comes from the website of the Norton Anthology of English Literature:

"The Gothic featured accounts of terrifying experiences in ancient castles — experiences connected with subterranean dungeons, secret passageways, flickering lamps, screams, moans, bloody hands, ghosts, graveyards, and the rest. By extension, it came to designate the macabre, mysterious, fantastic, supernatural, and, again, the terrifying, especially the pleasurably terrifying, in literature more generally."

So, how might weather in Jane Eyre relate to "the Gothic element" in the novel? From the first chapter of the book, the weather sets the mood: 

"Folds of scarlet drapery shut in my view to the right hand; to the left were the clear panes of glass, protecting, but not separating me from the drear November day.  At intervals, while turning over the leaves of my book, I studied the aspect of that winter afternoon.  Afar, it offered a pale blank of mist and cloud; near a scene of wet lawn and storm-beat shrub, with ceaseless rain sweeping away wildly before a long and lamentable blast" (Jane Eyre, Chapter 1)

In this passage, the weather not only mirrors Jane's sad mood, but sets the stage for the macabre/terrifying events that happen in Mrs. Reed's house: the red room, John Reed's abuse, etc. Similarly, in Chapter 23, storms and lighting add a sinister undercurrent to Rochester's proposal to Jane:

"But joy soon effaced every other feeling; and loud as the wind blew, near and deep as the thunder crashed, fierce and frequent as the lightning gleamed, cataract-like as the rain fell during a storm of two hours’ duration, I experienced no fear and little awe" (Chapter 23)

Although Jane is happy in this scene--she has just become engaged--the weather suggests that her happiness is precarious. It sets the stage for uncanny and scary things to continue to happen in the novel: for example, Bertha tearing Jane's veil in half in Chapter 25. Just before Jane describes that event to Rochester, the weather is tumultuous and threatening:

"I sought the orchard, driven to its shelter by the wind, which all day had blown strong and full from the south, without, however, bringing a speck of rain.  Instead of subsiding as night drew on, it seemed to augment its rush and deepen its roar: the trees blew steadfastly one way, never writhing round, and scarcely tossing back their boughs once in an hour; so continuous was the strain bending their branchy heads northward—the clouds drifted from pole to pole, fast following, mass on mass: no glimpse of blue sky had been visible that July day."

In short, the weather is one of many devices Brontë uses to establish the Gothic mood in the novel: terrifying, strange, and tumultuous weather mirrors the terrifying, strange, and tumultuous events that take place over the course of Jane's journey. Just as telling a ghost story on a dreary, rainy night makes the story more powerful and creepy, Brontë's use of windy, rainy, and desolate weather makes Jane Eyre more vivid and uncanny.