I'm aware of the velocity of money and the perception that poor people pour money back into the economy rapidly while rich people don't (and it matches my personal biases, so I like the idea a lot) but -- I'm trying to figure out what the rich can do with money that actually takes it out of the economy. Unless they actually stick dollars in suitcases and store them in the wine cellar, almost anything else I can think of puts the money in someone else's pocket one way or another. Stuck in banks: used to back bank loans.
You're starting to get it. It isn't whether poor people or rich people get more money - that's a fantasy concocted by people to justify their biases (whether it be dislike of rich people or dislike of poor people). What matters is the usefulness of the things they spend it on. For the money to improve the economy, it has to increase overall productivity. A rich guy spending money on gold-plated toilet seats doesn't increase productivity. A poor guy spending it on meth doesn't increase productivity. A rich guy spending it on a new computer which helps him organize his work and thus get more work done increases productivity. A poor guy spending it on food so his teenage daughter can stay home studying for high school instead of working so the family can eat increases productivity.
Kicking it up an abstraction level, the true currency is productivity. Money is just a (loosely tied) representation of productivity, distorted by inflation, inequality in negotiating leverage, corruption, etc. Things which increase people's productivity are good for the economy. Things which don't increase productivity or decrease it are bad for the economy. Whether raising the minimum wage helps or hurts the economy depends entirely on how it affects productivity.
Rich people are richer because they on average tend to spend their money more on things which increase their or others' productivity. Poor people are poorer because on average they tend to spend money more on things which don't improve their or other people's productivity as much. Are you angry and ready to hit reply because you think I'm saying poor people make stupid spending decisions and deserve to be poor? Then you need to re-read the previous paragraph. What I'm saying is if you guide and teach poor people how to spend their money more productively, they will become rich people. This isn't a zero-sum game - everyone can become rich if everyone makes good decisions. That is much more important than raising the minimum wage.