It all depends on the router and it functionality to be frankly honest. Some router may allow you to catch traffic and then it be stored on the router it self and all u have to do is grab the file then analysis it. The likely chance that this will be capable from home router is very limited to be honest.
If you can some how change some of the info on the router like the default DNS server address distributed via DHCP. You can get request to come to one of your own recursive DNS server. This will allow you to be able to monitor all the websites visited by client's on the network that get the DNS server info dynamically.
If you can somehow set-up a mandatory hop that is a must pass for all packets then i guess u can sniff all traffic. You would also need to set up a high performance server that forwards the traffic for you, this server would be the hop. I don't know whether this is possible on normal home router but i know you can set mandatory hops via QoS.
Thanks for the informative post. It makes sense and yes I was just not sure if it could be done remotely. These were not default logins by the way. Thanks for the info guys.