I can see it taking off, especially after you read articles like this
Wi-fi set to re-wire social rules
But only if the apps work on both the portable and the home PC without any modifications, this is one of the reasons I think that PDA's never became more popular, people simply don't want different platforms, and lets face it, most of the useful apps you use on a home based PC are never available on a PDA.
Wouldn't it be nice to carry your Home PC around in your pocket (albeit a large pocket), and be able to run the same apps that you do at home.
OK so high end games might not work and very processor intensive apps would also be a no no, but Web browsing (Obviously), VOIP (I would buy one for this alone, it would pay it's way in no time at all!), being able to hook it up to your digital camera, select and edit your pic and send it all from one device, whilst out and about. There's a million and one uses for such a device.
How about come up with some creative uses, I like the idea of the interactive game as mentioned on the BBC article above, and watching movies whilst travelling is another good use, imagine being late for work, stuck on the train, and being able to deal with all your e-mail before you get into the office etc.
I know, we've had laptops for years, but, they are still just that bit too big to make them ultra portable.