Well, the first thing that springs to mind is of course that gravity can theoretically affect all kinds of normal matter, including air molecules (wind) and of course electrons (lightning). Temperature changes within a gas can be achieved by compressing/expanding that gas, and this gravity can do nicely as well.
The second is the reverse correlation: that enemies are locked down by miniature weather patterns swirling so tightly around their bodies that they cannot move (immobs/holds), that highly focused, gale-force winds can throw stuff (propel, wormhole), and that great concentrations of miniature weather patterns can form a remote operating point from which the controller can call upon additional control (singularity).
Of course, those are just the first two that spring to my mind. There are probably many more rationales. 
"If I had Force powers, vacuum or not my cape/clothes/hair would always be blowing in the Dramatic Wind." - Tenzhi
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Now I just need a name. Heh!
That was a very good rationale, and I can't thank you enough.
I'm having a hard time finding a correlation between the two. I want to find some common link, or some info regarding gravity's effect on storms. Can anybody give me a hint?
[Btw: I know there is a link between swelling tides and gravity, but that isn't really strong enough for me to explain a thunderstorm.]