I highly doubt I would ever want this. It's kind of like Mint, which I almost never want, but worse. All of the cards I have in my current hand I either want to play this turn (the good cards), or don't want more of (the bad cards). Of course, there are a few exceptions -- gaining green in the end game, can make terminal collisions not as bad, etc. -- but I don't think these would counteract the drawbacks.
EDIT: ah, didn't even see the +4 cards. Ok, maybe I'd want it. Still not sure though. Regardless, I'd much rather just have an Ironworks and a Smithy in my hand than a Broker.
EDIT2: you'd need a way to determine the "random" card. Maybe some wording like "the player to your left chooses a card from your hand" (they only see the backs of the cards), or something like that.
EDIT3 (okay, last edit): I guess my main turn-off here is that I don't care for things in Dominion that make it more luck-based. Of course there is always shuffle luck, and that's necessary to make the game fun. But I'd prefer the game stay as skill-based as possible beyond that. Just my preference though.