My home broadband router has a dns service capability.
My question is, do I need to enable this for resolving IP addresses on my home network?

dimitridemetrius,

If the router is what various home internet providers might describe as a "residential gateway", and if you wish to have fixed local IP's for different machines at home, it may have the ability to provide that via DHCP. If your goal is broader, figuratively and literally, such as to allow access to some or all of the machines at home, from outside your local home network, then you can often specify which machine you wish to route to, depending on various factors. If you could give us some idea of what your goal is, we might be able to help more. In other words, in what context do you wish to "resolve IP addresses on your home network"?

As an example, to make it simple for me to reach one machine in my home network from another, some of them have been assigned fixed local IP addresses, but they get them via DHCP. Beyond that, I use external DNS, of the free type, to target the IP address assigned to the "residential gateway", but with a specific port. I then have the gateway route communication with the port to an associated standard port of a specific machine on the local network.

No. You can run a DNS server on other devices.

And if all that makes your head spin, the simple answer is likely "Yes". Connect any new device and it will get a local IP address and be able to connect out to the internet. If that is all you need for now, that's all you need to do. Additional features as above can be added later.
I don't think I've seen a ISP router (gateway) that isn't a DHCP server by default.

Sorry - you said DNS, I read DHCP ...

Welcome to LinuxQuestions.

Some home routers do have local DNS capabilities and without additional information I don't know if this is what settings you are looking at. Try it and see. DHCP address will be automatically added to allow you to use hostnames versus IP addressing to connect to other network devices.

I agree that normally part of a dhcp offer is also one that includes dns.

One can easily modify locally to force any of a number of dns servers but there could be some order to how a FQDN is resolved. Hosts files and things like proxy.pac can affect all that.

"Your mileage may vary,™" but my home router does offer DNS services.

Its DHCP implementation also allows you to assign predictable internal IP-addresses to devices based on their "MAC address."

However – the /etc/hosts file gives you the ability to resolve names locally. So, you could set up your router to assign known IP's to certain devices, then list them in the hosts-file.