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:
https://i.imgur.com/ThsYRPj.png
In Game:
https://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.
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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!
Wait for the file to get approved and download it. After that do the following;
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.
If you've got problems or questions, shoot me a message and I'll take care of it when I wake up. It's almost 5am, so I am going to get some rest.