While the Core i3 would be an upgrade from an E8400, it wouldn't be a huge upgrade and it has a higher power draw. And obviously it'd cost a lot more to do a whole rebuild now!
Intel and AMD both have new architectures coming out next year (for AMD it's the first major change since Athlon 64!) which should offer better performance with higher efficiency, but at the minute the cost of completely rebuilding your PC is not going to be translated into a noticable gain in performance, and is likely to leave you with a more power hungry PC than sticking with your E8400 + P5K and swapping the 8800GTX for a more efficient graphics card. From your OP it looks like saving money is high on your list of things to do, so surely that starts with reusing as much of the current rig as possible
If you went for a HD6850 now to replace the 8800GTX you'll get a noticable performance increase, a noticeable reduction in power draw, and then if next year's new CPUs turn out to be a lot more energy efficient than an E8400 you could replace the rest of the PC then, carrying the 6850 over to an even more efficient build on a new architecture!
EDIT: have just looked up some reviews and it appears that you might save
a few watts by going Core i3, but
tbh I don't think that's a big enough difference to justify the £300 outlay up-front that it would cost to make the change, and the performance (as I said earlier) is likely to be no better except in heavily threaded workloads....