Like most routers these days the UniFi edgerouters have convenient support for feeding the local IP address into a dynamicDNS provider. Due to a mismatch between how things are labled in the router settings and NameCheap it is not at all clear how to set this up. This is further compounded by there being no feedback from the router that it has done anything or that there is an error.
You need four pieces of information for the UniFi configuration the first of these is your Dynamic DNS password.
In the “Domain Details” on NameCheap at the bottom of the AdvancedDNS page you will find DynamicDNS settings. Enable it, then note the “Dynamic DNS Password” which is a generated string and different from your NameCheap login.
In the UniFi controller, select System/Dynamic DNS. Fill out the fields as follows [Examples given for myhouse.mydomain.com]:
So… pretty unobvious. Documented for posterity and my future forgetful self.