I say get a good body 66 mustang and put a monster V-8 in it. I am a ford guy so just throwing that out there. If it is only going to be used for a weekend car then that is what I would get. I like the Honda S2000 but they are showing their age now. If you are looking for a track car then the Miata and S2000 is the obvious choice. Classic cars will never be able to get the handling you can get from those cars. Speed yes, turning no...
If you are looking for a fun mess around car then I think classic is the way to go. Apperance is timeless and many have kits now give the ability to upgrade to newer technologies(ie disc brakes, fuel injection, and so on) Just my 2 cents.
If it were me I would not buy a junker and rebuild it. I would buy a very good condition one. Let someone else deal with all the little issues that come up while rebuilding a car. They can be had very cheap if you don't go with an all original car. Anything all original is going to cost some serious coin.
Good luck.