>1) What would be special about it?
It should be so simple to use that a computer-challenged individual could use it, yet powerful enough for advanced endeavors. A compact design that doesn't clutter; something like the MacMini or iMac, or the A500|A600|A1200 line.
>What would make people want to buy it instead of a PC?
In America, competitive in price, which is now about $395 for a base system. It would have to be focused on the consumer market. A Productivity Application Suite would need to be able to read and write in MS-Office datatype formats.
>2) How can the Amiga recreate the charm it once had as an inexpensive multimedia computer?
Cost $395 and be able to do word processing, spreadsheets, email communications, save and organize pictures easily, databases, acccounting, JavaScript, Java, Acrobat Reader, Flash, Shockwave, and some good quality games which Amiga has a good collection though some are outdated.
>3) What are the hardware solutions for the Amiga considering it typically uses hardware that can be outdated, limited in quantity, generally incompatible with market leader?
Something that's cheap, readily available, will be around for a long time, and designed with efficiency as top priority. In reality, Amiga needs a technology Partner, and it would be an x86 manufacture. Even Sun Microsystems ported their SolarisOS to x86, so that should be motivation enough business-wise.
In brief, use low cost hardware (x86, get over it cause there's nothing cheaper), port AmigaOS4 to it, create and include in the base system aLife and aWork (similar to iLife and iWork) application suites, and include Firefox as a modern web browser.
Then market the computer to China, because they have plenty of people to buy and to help develop for it.

Actually, working with a company from Japan may not be a bad idea because they have gadgets and technologies that are more advanced, technical, and "gadgety" than America.