Which random money schemes do you think would be best at this?
Bitcoin, interestingly enough. Bitcoin has multiple competing blockchain lotteries which are often provably fair, and which generally take <2% in fees, while offering high-leverage bets; for example, SatoshiDice famously paid out 1920btc on a 0.03 wager.
These are using cryptographic hashes, which are unpredictable but deterministic, so they're not immediately applicable to MWI. What you could do for MWI purposes is use a quantum RNG to choose at what point in a day (or month) you bet at; since each block gives you a different chance at winning, quantumly choosing between enough blocks can essentially guarantee a win for at least one branch.
So, assuming you had the necessary bitcoins on hand, this is actually a very easy way to leverage up (far easier than some suggestions I've seen like trading foreign exchange options). Run the RNG, send it whenever, and within 10-15 minutes you'll have won a large fortune or lost a small one for a fee of ~1.8% of your bet.
I was actually thinking about this as a strategy for myself. I have a decent number of bitcoins (I began accumulating what I could back in May), but I don't have a life-changing number of bitcoins. On the other hand, if I bet at, say, 10:1 odds on SatoshiDice, that combined with future bitcoin price increases would be life-changing. I could fire up a quantum RNG, ask it when to bet over the next week, bet, say 25btc, and in one of 10 universes, walk away with ~248btc and cash out for $200k.
Is this a great idea or a terrible idea? I really don't know.
I wouldn't suggest anyone cash out all of their bitcoins... rebalance them instead. I'm actually surprised I haven't heard about any wealthy people/hedge funds/etc. using volatility harvesting strategies to trade bitcoins, their volatility is so insanely high it seems like an obvious strategy.
I haven't been able to find the source of the idea, but I've recently been reminded of:
This is, of course, based on the Multiple Worlds Interpretation: if the lottery has one-in-a-million odds, then for every million timelines in which you buy a lottery ticket, in one timeline you'll win it. There's a certain amount of friction - it's not a perfect wealth transfer - based on the lottery's odds. But, looked at from this perspective, the question of "should I buy a lottery ticket?" seems like it might be slightly more complicated than "it's a tax on idiots".
But I'm reminded of my current .sig: "Then again, I could be wrong." And even if this is, in fact, a valid viewpoint, it brings up further questions, such as: how can the friction be minimized, and the efficiency of the transfer be maximized? Does deliberately introducing randomness at any point in the process ensure that at least some of your MWI-selves gain a benefit, as opposed to buying a ticket after the numbers have been chosen but before they've been revealed?
How interesting can this idea be made to be?