Letting users define their own nickname on the node.dockerapp.io subdomain is quite difficult as every customer would share the same common subdomain, leading to name conflicts and security problems (you could “probe” if a nickname is available, and if not, you know that someone has a node behind and you could DDOS it)
What you can do, if you have bought a domain name, to add an entry in your DNS records of type “CNAME” that points to the docker FQDN.
For example, you have bought mycompany.com, you add a CNAME record “mygreatname” pointing to 518fc1a0-828d-4d5a-9b54-2b9d0062fdc5.node.dockerapp.io. (the dot at the end, after the “io”, is important in such case).
Your node will be available as mygreatname.mycompany.com a few minutes/hours later (due to DNS propagation).