Having done no research on this (aka. Someone else probably brought up what I'm about to say and proved/disproved it), I bet you could run a few AM Block Erupter USB sticks on most routers that let you pull open a shell and have enough resources to install/compile and run bfg/cgminer.
I doubt there's enough interest in the general public for the manufacturers of home routers to make bitcoin mining a built-in function of the devices but hosting the AM sticks seems like a doable task to achieve what you're saying.
Definately doable. Many routers these days actually run Linux. Using certain NAS units also would be a great option. I believe there even are some that allow you to install software.
1) You use a router anyways.
Yeah but it is already provided by the ISP... isn't it in your case?
Not in my case. Depends on the ISP. Most ADSL/VDSL/SDSL providers out here do provide a modem-router box, but Cable internet usually doesn't provide a router (or only at additional cost), nor do glassfiber ISP's.
Especially glassfiber is usually only with a glassfiber-to-RJ45 terminationpoint. From that point on is your own responsibility.
I think OP would be talking about a third party router. Probably custom built.
I don't think it's a bad idea. I just don't like mixing my platforms/systems/etc. I like my routers tight.
So you own a PC for every different task too?
I have no problems with mixing things. It'll only result in a lower electricity bill.
Why not combine a NAS, Miner, Router, unmanaged downloadbox, mediasharing (for those TV boxes), etc into one machine?
Single point of failure? yes. But is that really a problem for most tech-savvy people? It's not like your life depends on it. If internet is really important, you could keep a router just as backup and connect it as soon as you need it.