Solar power backup for router
1.) Golf Cart batteries are the easiest supply for a home electric backup battery bank. 

2.) Bigger is better with the solar panel array. If you're only going to be running the router (why not the modem too? Or is it combined?) you can probably get away with a cheap small solar kit from these guys : Silicon Solar They offer some great battery charger kits that pretty much take all the work out of it for you. You'd still need to wire these to your battery and convert the current to your AC plug for the router. 

3.) For connecting the Router to the power grid you've created in Steps 1&amp;2 you'll need an inverter. Depending on your interest in making something pretty to look at and safe to handle you might want to do some more research on an inverter you can wire to your power grid. 

Here's a graphic that should help explain the setup and get you thinking visually on the setup explained above:



In your case the panel box would be replaced with the inverter output device combined as one item since you're only looking to power 1 device.