I've been reading the 'Hackers' book by Steven Levy, and the way it describes what makes hackers what they are is exactly this kind of tinkering and problem solving.
One thing that's been fascinating me this past decade or so is that clearly programming and lots of other 'nerdy' stuff is quite mainstream now, and as a result the 'old-fashioned' concept of a hacker (or the 'hacker ethos' or whatever) seems to have become less of a specific subculture.
But at the same time, I've been trying to do some 'code school' style teaching and I find that there's still a very distinct difference between the coders who mostly just want a job and see coding as a viable path (and even enjoy it, much of the time), and coders who have that same kind of 'hacker' mentality.
It's hard to describe, because it's not just about skill or 'good code' or 'productive results'. I guess it's some degree of unusual passion or excitement about the whole thing. And definitely a tendency to just 'hack it' and coming up with sometimes horrifyingly creative solutions to the problem.
For example, one guy I was teaching was incredibly messy in his code style, undisciplined, and clearly copy/pasting stuff from Stack Overflow on occasion. But when we talked about computers, programming, what kind of cool stuff he'd like to build, he was clearly different from the others. Where the others would sometimes diligently go through the Rails Tutorial chapter by chapter, with him sometimes the only thing that got him to actually learn was to give him something that 1) tickled his particular passions (games), and 2) was problematic enough that he'd be cursing at his computer until well after the 'official' study time.
And while his solution rather predictably was often messy, different from what he'd been taught, and somewhat unconventional, he usually figured it out and there was almost always some 'cleverness' to it.