So this New Year's Eve, I set out to solve this problem and this is the result, ICanHasNoise, a web service that lets you generate background images with noise with just a URL. For example, the URL http://icanhasnoise.com/123456/200x200/4 would generate a noisy background with color #123456 (gradient backgrounds are also supported), size 200x200 pixels, and with a 'noisiness' value of 4. This URL can then be slotted into to a CSS background-image rule and voilà, your element is noisily backgrounded without the involvement of any sort of image editor, and if you want to tweak the background, all you have now to do is change the URL.
I built this website to scratch my own itch and I hope that others will find the site useful too. Thanks for even bothering to take a look and as always, feedback is very much appreciated!