The blinking Internet light just indicates traffic. If you're having difficulty accessing things like Online games, but Facebook and such work, make sure the router's firewall isn't turned up too high. If you haven't made extensive changes to the router, it wouldn't hurt to give it a factory reset just to rule out anything with the router. It'll go back to the default configuration it came with when Verizon installed FiOS or when you installed the router.
Now if that Internet light is blinking rapidly at all times of the day and night, and you've got devices turned off, check your cabling. Make sure the router isn't plugged into itself anywhere.
