Have a look here if you aren't aware of it.
It's one of those concepts that I've seen in many games I've been a part of, but nothing particularly formal as a genre. It's the way we always played Shadowrun back in the day, sleazy overnight hotels, or sleeping in the back of the car when times were tough...trying to stay honourable, when the dishonourable thing might earn enough money for a good meal or repairs to equipment...not cashing in too many favours now, because you never know what might lie just around the corner.
25 years later, it's not Shadowrun...we just call it life. I've been in this place, I'm not quite there now thanks to a halfway decent social security network in Australia, and as a university student (and carer for a chronically unwell wife), living in a house rented from a friend, I'm afforded enough welfare to generally cover rent, pay most of the bills, pay for enough food, cover car repayment to my parents, and occasionally splurge on something special. Income from game design and freelance work covers the occasional luxury, like more gaming products, a pair of concert tickets, or a good night out. But without the social security safety net, this is exactly where I'd be.
I know plenty of people who'll never live the 'Rentpunk' experience, people with family who give them houses (or interest free long term loans on houses that will never have to be paid back because the houses are in their parent's will), "loan" them cars, arrange jobs for them at prestigious companies where they don't have to do a lot but get paid more than the people who do perform the actual work necessary for the company's survival...such people grow up in the right circles to get "gallery openings" for their work if they choose to follow a non-conventional career path in the arts...no matter what path they take, they have an intrinsic degree of privilege that they assume the rest of the world also has. They think nothing of it, and actually thinking they are working hard to get noticed, but these are people cushioned from the 'Rentpunk' world...society works for them (mostly because they don't actually have to deal with it).
In a healthy culture, the cream rises to the top...in an unhealthy culture the scum sits on the surface. Naturally, those on the top want to say that they are the cream, the elite. They want to say that society is healthy, and they are it's paragons...those who make up the bulk of society may strive to be like them, to gain their place near the top (not realising how loaded the system is against them)...they may choose to completely rebel against the system...or they may try to walk the delicate path between extremes, trying to show the hypocrisy off those at the top, while trying to invigorate a new society without the need for complete nihilistic revolution.
My interpretation of 'Rentpunk' would keep these things in mind, ideals that the lowest rungs of society cling to in the hope to get out of their miserable cycle. Do you give up your morals to ascend, and become the very thing that you hate, thus perpetuating the cycle onto a new generation? Do you give up your chance at climbing to the top of this society (and if so, do you choose to forge a new society)?
The whole thing could easily be a game driven by the 'FUBAR' engine...Do you succeed? Do you fail? Do you have to give up anything in the attempt? Then...how does the result of the action attempt influence you? And, how does the result influence the outside world?
I could apply these concepts to the steampunk-pirate setting I developed last year, perhaps a game focusing on the pirates, privateers and settlers living in the slums of the main towns...or maybe revisit the mutant animal game I started last year. Or maybe attach it to one of the dozen other demi-concepts I've considered over the years.