For a while I was using a mod called Lol Enhancement Suite to do this. It patched in an option to randomize your selected skin, and then once you got to the loading screen you could see which one it picked. It was great, but the only problem was that every time League got patched, I had to re-run the mod. Certainly became a bit of a nuisance. Plus, in the more recent versions of the mod, they've disabled this feature altogether, so I can no longer patch in a random skin button.
As a result, it would be great to have this feature integrated into the client. I have so many skins from mystery gifts and general purchases, and I'd love to be able to give all of them the love they deserve, not just the most recent or expensive one. It wouldn't have to be anything fancy. Just an option on the skin selection screen labelled "Random Skin" or something similar.
Here's hoping that Riot sees this and considers implementing it! :)