Boy, is this going to be an interesting thread...
Of course the flippant (and perhaps wise) answer is to save everything. However, that's often not practical...(and having said that, I
KNOW we've all got stashes of parts laying around.
So, my list:
1) Trim bits - anything small. Much of this can be purchased, but some cannot. Things like badges, chrome trim, grilles, headlamp trim rings, aluminum cockpit surrounds, etc.
2) Electrical - Lucas stuff is good to keep (small and easily stored). Certainly instruments, switches, etc.
3) Front disc brakes - if you have a spare set, save everything from the A-arm out. Folks (like me) are looking to buy these to convert Bugeyes from front drums to discs (hint, hint--PM me if you're interested in selling these parts

)
4) Rear axle (if you have room). If not, certainly save the gear cluster, and especially the half-shafts (which tend to break).
5) Windshield (complex curve makes replacement expensive--wish the movers hadn't broken my spare

)
Less critical, but still worth saving (if you have the space):
1) Body panels (rust free) - always worth saving
2) Major mechanicals - engine, gearbox. However, the engines are plentiful (but never hurts to have a good "core") and gearboxes are rapidly becoming worthless (as folks are putting in Rivergate conversions, leaving the smoothcase/ribcase trannys for the concours crowd).
3) Interior - seats, trim panels - likely to be replaced in most restorations, but worth saving for patterns.
4) Steering rack
5) Radiator
6) Generator/Starter (if you have a "tach drive" generator, that goes in the "must save" category (along with the little tach-drive gearbox), but you mentioned a '66 so I believe you have an electrical tach)
7) Wheels/Rims (especially if you have Bugeye wheels--the ones with "holes" around the circumference)
Anyone else? Anything important that I've forgotten?
d