That would be a fuel solenoid. In most cases the purpose is to shut off fuel to the carb. when the ignition switch is turned off. If the float should stick open, for instance, no fuel can enter from the line to flood the engine or the floor.
Since I have not tested one on an LX188, I'll just give the generic test. With the parking brake engaged, and when the key is turned on, there should be an audible click at the solenoid. You might have to have someone in the seat too, not sure about that. You should probably be able to feel it if you can put a finger on it and still reach the key.
If you have a test light or volt meter, you can check for voltage at the wire end with key on. There is a little plunger inside that retracts as voltage is applied. Most are spring loaded to return to a tapered seat when voltage is removed. Guess it could be gummed up and sticking. Or a bad connection could make it work intermittently. A new one is... $75 bucks, gulp!
Take it off, clean it, and see if it's working correctly. I'll try to find more info for you.