For the purposes of this question, "random" will include pseudorandom and quasirandom alternatives.
This page allows you to create a pseudorandom image of size 256 by 256, determined by the image name you specify. (See here for a brief explanation.) It's a slow process taking about 1 minute. In theory one could make a larger image from a collage (although the images might not look good together), but my experiments with multitasking across tabs indicate this saves no time due to processes in idle tabs pausing.
What I'm looking for instead is a quick way of creating a single larger image, be it of custom or standardised size (my use case seeks 1600 by 2400). I'm happy to use a webpage, downloadable program, or script written in any common programming language.