I like the idea of using the flushing handle to activate the auto closing mechanism. However, I'm not sure if using a magnet is the best option. If you are thinking of using an electromagnet, then it will need an electrical source. This adds some unnecessary complexity and possibly a safety hazard. Water and electricity don't play together well.

You might be able to power the seat closing mechanism by using the water. When you flush a typical toilet, some of the fresh water is sent through a tube that delivers it directly to the bowl. I'm talking about that thin plastic tube that you see on top of the big vertical pipe just under the tank cover. You could get a longer tube and route it to a piston mounted to the front of the tank behind the raised seat. When the toilet is flushed, water runs through the tube which pushes the piston forward. This pushes the top of the seat forward, causing it to close. There are already seats designed to drop slowly available at most major hardware stores.

The closing device could be installed by simply hanging it on the front of the tank. The tank cover would sit on top of it to hold it in place. You would run a tube from the valve to the device and another tube back to the pipe so that the water could still go to the bowl as usual. The actuator could use a piston directly or a lever arranged to push the seat forward.

This might be a fun project to experiment with, though I don't personally need one. I always close the lid on my toilet before I flush it. (BTW, I just designed this thing in my head as I was typing. Am I good or what?)

Kohler (and maybe others) already makes a toilet seat that flushes the toilet when you put the lid down...sort of the same idea in reverse. You have to put the seat down to flush it.

TOTO toilets