Step 1: Parts
Wind cups, part #7903 from Davis Instruments (www.davisnet.com), cost around $15.
Bike speedometer from any bike shop
2 ball bearings, 1/8" ID from any hobby shop (used in radio control models)
Once you have the parts, this project should take less than three hours to construct I think.
Notes on the parts:
For the Anemometer head we use the "wind cups" from Davis Instruments. This is a significant time-saver over constructing your own wind cups! Davis sells anemometers and weather stations, their complete devices cost around $200 i think. So if you'd rather just buy a nice anemometer, i'd highly recommend theirs. The wind cups we are using are considered "replacement parts", you might need to tell them you have one of their weather station products but you broke the wind cups. I don't think they really care though. Anyway, the wind cups are a nice plastic assembly, about 6 inches in diameter with three cups attached to a central hub. The hub has a 1/8 inch hole for a shaft, and it has a magnet embedded in it for use with a magnetic reed switch for detecting rotation. They have a photo on-line of their entire wind-assembly, the wind-cups you'll be getting are just the piece at the bottom of this assembly.
In a future version of this project I hope to make my own wind cups. I've tried a couple designs out of cut-up beer cans and soda bottles, but have not gotten anything that works well enough...
It's quite convenient to use a bike speedometer here, because they detect bike speed using a magnetic reed switch. On a bike, you attach a magnet to one of your front wheel spokes and then you attach the reed switch to the front fork. Every time the magnet on the spinning wheel passes the reed switch, the switch pops closed and then open again. The speedometer detects the open-close-open of the switch, and speed is computed by how often the switch is activated.