Solar-power backed router
I know you've specified that you want to use solar panels, but have you considered using a UPS?

A quick search comes up with a $115 550VA/330W APC UPS which should be adequate to supply a router for a few hours at least, possibly more.

This will also be cheaper than providing the equivalent power in solar panels, be more reliable (doesn't rely on a sunny day), and far easier to set up.