I'd imagine with no minimum weight, lightening the car would become a bigger priority, perhaps as important as aero, and more development and innovation would happen in the area. The way to arrange cheap, standard composite materials into ever better and better structures whilst exceeding the crash test requirements, could provide some advancements in race car construction... After all composite formula car chassis are nearing 30 year-old tech and perhaps further innovation and excellence is possible if it was top priority? 
There would be a shift, eventually. But first, the teams would have to adapt to the new tyre and aero environment. Because if you left the formula as it is now, but removed the minimum weight, the cars would be faster around corners that the human body could sustain - the drivers would pass out.
But if you slowed the cars down by reducing aero and tyres, weight would then become a major priority. But do not think that it is not now - the CofG of F1 cars is a major issue, as is the polar moment. Its just that the huge cost of lowering CofG is not as rewarding as spending money on other things. While Honda developed the first good all carbon gearbox casing (with a bit of titanium only used), most teams kept their titanium / carbon boxes. Williams stuck with aluminium - it cost them less than a kg, but saved them half a million pounds. This year there has been a major weight issue though, due to the weight of KERS. It was weight ballance that likely hurt KERS. McLaren claim their system does not hurt the weight balance. And it seems their car handles well, even with the KERS installed. Weight still matters now. With the relative increase in front tyre contact patch, teams needed to shift the weight further forward this year. Hence RBR - who did not have a KERS weight issue because they did not envisage using it - built an almost all carbon gearbox casing, which was very expensive but it was lighter and it allowed some extra ballast to be shifted forward in the car.
Weight still is vital - its just that due to ballast, the rewards of light weight parts are the centre of gravity benefits only, and the different polar moment benefits (having less weight at the ends of the car allows the car to turn more easily).
Another idea might be to ban tungston, and to stipulate that only cast iron was allowed as ballast. That would put much more emphasis on the upper body weight IMO.
Edited by Melbourne Park, 27 July 2009 - 23:54.