boogerhead ,
As for the minimum it only has to be as much as the Fios Router is rated for and your internet provider internet speed. More than likely cat5e is plenty. For future you could also install a cat6 or cat6a cable.
If you have never worked with Ethernet cable and installed jacks and or plugs I would recommend you buy a factory made, (patch cord), cable with RJ45 modular plugs installed. Just make sure you protect the end that you will be pulling from point A to point B. Also be careful installing the cable. Avoid any sharp bends or kinking of the cable. If the cable is kinked straightening out the kink will not restore the twisted pair lay of the cable pairs. If kinked, the cable is damaged. Kinks can cause errors and slow down the speed. Also do not tightly coil up any excess cable at either end. This can also cause errors.
I would recommend you buy the cable from Blue Jeans Cable.
https://www.bluejeanscable.com/store/data-cables/index.htm.