The skinny wires likely provide power to the primary side of the relay. Look that term up, but it only needs to be robust enough to power the indicator LED on the switch and the relay.
Using a different switch wont be difficult. Switches that have an LED indicator have 3 wires. One is from the battery, one goes to the relay to switch it on, and the last is a ground. The ground is what allows the indicator LED to illuminate when you switch the lights on.
I'm pretty sure the Toyota switch uses a spade connector, it should look like this.
Go to the parts store and get spade connectors that will slide over the metal contacts on the switch.