In Shakespeare's Macbeth, the "Thunder and lightning" in the opening stage directions of the play don't, in themselves, "represent" anything. In other words, they're not symbols in themselves. They are more indicative of other aspects of the play; they contribute to other aspects.
Together with the witches, the weather creates an eerie atmosphere that will pervade the play, and the First Witch, of course, references the weather, and particularly the weather to come, in her lines that open the play:
When shall we three meet again?
In thunder, lightning, or in rain?
This draws attention to the weather, and the weather is, or at least soon will be, highly suggestive. The day is stormy, as is the state of affairs in Scotland. The witches close the opening seen by again referencing the weather, but also by connecting the foul weather with the political affairs in Scotland, though this is only implied by this point:
Fair is foul, and foul is fair,
Hover through the fog and filthy air.
Macbeth unkowingly mimics the witches, due to the weather, in Act 1.3.39 when he enters the stage with Banquo:
So foul and fair a day I have not seen.
This connects Macbeth to the witches, and foreshadows things to come.
During the course of the play, foul weather will contribute to the multiple meanings of the fair and foul statements (thanes that are assumed to be loyal are not, Lady Macbeth wants to be a man, Malcolm suspects Macduff of being a traitor when he is not, etc.), and become indicative of the unnatural way in which Macbeth gains power and keeps it.
The weather, then, sets the stage for things to come. It contributes to theme and is indicative of the political situation in Scotland. It's difficult to see the witches meeting Macbeth and serving as catalysts for the murder of a king, the murder of Banquo, and the slaughter of Macduff's family, in bright, cheery sunshine.
arrival of the witches