Riot specifically added a picking mode that allows this a while ago; I'm surprised that it's remained more or less undiscovered for this long. Internally, this mode is called GAME_CFG_BLIND_DUPE. While there's no way to activate this mode with the live client, it's quite easy as a developer (or if you simply mod the live client).
The mode works mostly as you'd expect: you can pick champions multiple times, and they're all playable in game. There are a few cool quirks though, like picking 10 Dravens and catching axes
Here's me using it:
Champion select: http://i.imgur.com/ThsYRPj.png
In Game: http://i.imgur.com/IC5PJo7.png
If you want to use this yourself, keep an eye out on Reddit (or, it might be easier to follow @astralfoxy). Sweet Stuff is arriving next week, and I think you'll freaking love what you see.
MOD FOR LIVE CLIENT
Your overwhelmingly positive response has worn off on me! In response I am presenting you with a quick modification so you can try this out right now. Remember, you use this modification at your own risk; you might get banned; the world might explode; the usual disclaimers apply. Riot may patch this incredibly quickly, so good luck and have fun while you can!
Copy it into C:\Riot Games\League of Legends\RADS\projects\lol_air_client\releases\0.0. 1.51\deploy\lib. You can back up the existing file if you wish, otherwise you may need to repair your client later.
Open the client and create a custom game. The new custom game will be a duplicate pick game regardless of your actual choice of mode (blind pick, aram, etc).
Only the host has to do this.
Note: You may run into bug where you can't restart a game after someone dodges in champion select. This is a long standing bug related to custom games that Riot hasn't fixed. It's got nothing to do with this modification.