My initial impulse is to pick a random cell which must not be zero, generate a random number for each other cell and a random non-zerp number for that one. I'm not immediately decided on whether it's uniformly distributed.
Any algorithm that cares about the number of non-zeros could have non-trivial interactions with their arrangement and count, so picking something that generates non-trivial sparsity (and doesn't just make the array look like white noise) is going to have the best chance of exposing interesting behavior. The tricky part is thinking through how to generate "interesting" patterns, which admittedly I haven't put enough thought into.
Right! And even if it were, in the sense that that's what we should expect as real world input, it wouldn't generally be the best distribution for finding bugs.