High frequency mains transmission
Yes, transformers could be lighter.  400 Hz power is used on aircraft for this reason.

The generator wouldn't need to spin faster if it is designed with more poles.  Motors would also have to be redesigned to run at normal speeds with 1000 Hz power.

But how often are 60 Hz transformers used in modern electronics?  There is a big one on the pole outside of my house, and in my boat-anchor hi-fi amps and tube guitar amps.  All the other power supplies in my house use switching supplies that run their transformers at a much higher frequency.

edit:  Another thing, transformers tend to vibrate slightly during operation.  A 60 Hz transformer will "hum" a little bit.  But imagine all the appliances in your house buzzing at 1 kHz.  1000 Hz is two octaves above middle C, and our ears are 30dB more sensitive to this frequency than 60 Hz:


Image source: http://en.wikipedia.org/wiki/Equal-loudness_contour