Lightning is coupled to rainy times because lightning occurs when there are lots of clouds around. Why does the lightning occur at all? Large air currents cause water molecules to rise quickly in large, dark clouds. As the water molecules rise and turn into ice, they bump into other water molecules and ice and thus generate static charges. What ends up happening is positive charge accumulates at the top of the cloud and negative charge accumulates at the bottom. Now, the ground has a combination of positive AND negative charges to have what is called an average of "0 potential" but since it is very, very large, it can absorb discharge from the cloud. When enough negative charges accumulate at the base of a cloud, the cloud discharges and we see lightning.

I don't think we know yet exactly what creates the separation of electrical charges that makes lightning possible, but it's been seen on Venus and Jupiter as well as Earth, neither of which has rain as we normally think of it. Lightning on Earth happens in towering clouds of great vertical extent, not in flatter rainclouds, which means that it probably has something to do with the updrafts that happen in such large clouds. Again, so far as I am aware, the true cause of lightning remains a mystery, but the fact that it happens on planets that have quite different atmospheres from the Earth can provide some clues.