Product review (5 stars): Fixed the washer for <$10
I didn't think you could buy these types of parts on Amazon, but this was a bargain. You can easily pay $25 online for this part elsewhere or even $80 for an appliance repair guy to come install this lid switch. Or you can just fix it yourself.

Our washer would fill, but then the cycle wouldn't start. There are a few things that can cause this, but frequently it is the lid switch. When we would lift up the lid, it wouldn't make the "clicking" sound that it normally would. This is a sign that the safety switch (this piece) might have gone out. This cuts the circuit to the motor when you lift the lid so that it won't keep dangerously spinning when the lid is open. We were able to just pull the panel off the washer, bridge the circuit with another wire to make sure that the switch really was out and that the spin cycle would start once we bridged the current, and then replace the switch by swapping the wires on the old and new one.

It's a pretty simple job--I had a friend who had recently worked on his own washer (nearly identical) help me, but it's an easy DIY job if you can diagnose it correctly. It was certainly cheaper than a new washing machine.