Having reflected on this thread for a while, I think I'm going to gently shove at it to see if I can't get it to move in a different direction. So, apologies in advance for the hijacking.
I think the Amiga definitely has a place in the modern world: to remind us of the poor quality of modern computers. It's certainly not alone in that place, either, but it's a good example, nonetheless.
The 'modern' PC is a piece of junk. It's designed to be (partially) backwards compatible with systems that no one in their right mind would _ever_ want to actually use. Of course, the PC architecture has changed significantly from the days of AT PCs, probably to the point where very little code from that era would actually run on a brand-new PC.
However, the staying power of the x86 platform is largely based on this idea that a PC built today should work with junky old code from 5 years ago. This, imho, is also the reason why PCs are garbage. It's not the cost cutting, the commodity grade components, no! It's the fact that there hasn't been a significant break from the design flaws of PCs built to be compatible with older flawed PCs, which in turn were compatible with even older junk boxes, and so on.
We need to just toss the PC platform into the waste bin, and move on. I don't mean we should ignore what we've learned as we went along, in fact, quite the opposite. As a society we are in _desperate_ need of reliable computers that can be programmed to do their jobs without it taking man-years of time to make it happen.
I was re-reading John Backus's seminal paper (from the Communications of the ACM, August 1978) the other day, and it occurred to me that as much as we need to break away from the von Neumann style of programming to accomplish this goal, we also need to a modern computer system built from the ground up in a style that discourages things symptomatic of the root problem that causes our systems to be unreliable.
Among these I would list code bloat in general, multi-megabyte device drivers, APIs that not only include the kitchen sink but many different implementations of one, and the lack of any mechanism for actually sharing libraries that aren't part of the basic functionality of the system. [I'm not talking about the technological capacity to link to them here; I mean the ability to actually derive income from the distribution of library code in a way that protects those of use who write it. Of late, I've been dreaming about an iTunes Store-like apparatus for selling software. I don't know that that would work, but it might just be worth a try.]
Obviously, the Amiga never solved all of these problems; nor have any of the other 'cool' systems that have come and gone. But if we look back at them with nostalgia now, it's because their design, and the philosophy of design that went with them, is relevant to our every day, 'modern', experiences with technology.
It's about time some of us took the lessons that can be learned from systems like the Amiga, and made the effort to build a computer that is beautiful, functional, and not hampered by the ridiculous assumption that someone really wants to run off the shelf software from a half decade ago or more. [As far as I can tell, the only software that fits _that_ description is the AmigaOS and a few other bits of technology that have similarly painted themselves into corners.]