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Right now I'm randomizing the position of my 2D rain's spawn positions on both the X and, slightly, on the Y axis. It looks pretty weird right now, it has a bunch of huge clumps and unfilled areas:

enter image description here

I want a more uniform, but not a completely boring and uniform distribution of my rain. I want something that doesn't look like a perfectly tiled texture, but doesn't look as crazy random as the picture above. What can I do?

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Divide your rectangular area where you want to generate particles into tiles then spawn particles randomly inside of those tiles (round robin randomly or some other method) rather than uniformly across the whole rectangle.

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