Creating a whitelist of websites on the router

Solution 1:

The easiest solution would be setting the DNS server (usually found in configuration settings of the router) to the one which does the needed filtering (one of examples could be OpenDNS, http://www.opendns.com/).

You could set up a similar thing on the router itself, but it is not so straightforward. You will need to install an alternate firmware (like OpenWRT, see http://wiki.openwrt.org/toh/gigaset/sl2-141-i) and then set up DNSMasq to only resolve on specified domains.

If you have a Raspberry Pi lying around, you could try if pi-hole can be configured with an allow list, instead of a deny list.

Of course these systems can be fooled if someone sets the DNS server on the computer itself, but while you let the child act only on a limited account that should be OK.