Fuck you money is a fancy way of saying enough money that you're completely financially independent.
In other words, you can say "fuck you" to an employer or a business proposition and it won't affect your living standards noticeably.
To try to put it in more financial terms, distinguishing between a stock and a flow, it means that you have enough liquid assets that they produce enough income from those assets to live on without depreciating or risking the principal. And maybe have some extra to splash on goodies (we're in fuck you territory here, after all).
In the United States, what's generally considered affluent is a family income of $250K/year, because after that if your tax burden changes it's "tax cuts for the rich."
That gives us a target. Right now (and for the foreseeable future), Treasuries yield 0%. But let's assume that a basket of bonds and dividend-paying blue-chip equities would be safe and could yield, say, 3%. (Maybe you could get more, and if so that's gravy, but let's be conservative here.)
So if you need enough money that yielding 3% will give you an income of $250K/year, that means you need $250K * 33.3 = $8,333,333.33. Let's round it to 10 so you can buy a house and a BMW (and maybe angel invest -- and don't be a dick, donate to charity).
So there you go: fuck you money is officially and scientifically 10 million dollars (after taxes, in the bank).
To anyone out there with stock options, get to your spreadsheets...