I'm also going to suggest an '80's-'90's MTB as a good tinkering project. Get one from one of the 'major' brands like Trek, Specialized, GT or Raleigh, Fuji or Nishiki for good build quality, and decent components.
Going late 80's early '90's will also ensure that you can find any replacement components new(er) rather than having to source vintage stuff for the '70s road machines show above. Think Small-Block Chevy vs. Jag inline-6
Early MTB's are also very versatile in what you can build with them, just by changing handlebars and tires.
See these threads from the 'Classic&Vintage' forum for some ideas.
Vintage MTB To Upright Bar / Urban Bike Conversions
Show Your Vintage MTB Drop Bar Conversions