Although if they added the "fill in my ip" button as someone else suggested, it might be useful for connecting to customer machines for debugging. Instead of "go to control panel, look for network connections, ...." etc. it would be just - go to ipq, click ok, give me the result. (also, short name might be easier to say over the phone than ip)
Being able to open up an Apache or Nginx server without using a confusingly long hostname (eg 127-0-0-1-dhcp.node01.someispdomain.net) which would be easy to misspell over the phone, to let someone grab whatever they needed directly from me makes this a very optimal service for such a small use.
I can see a group of people working on a webapp building a list of subdomains using this service so that a project lead can instantly see how each developer is working on a problem on their local machines.
The downside I see here is that some (many?) home routers support DynDNS service to automatically update your IP address with DynDNS whenever it changes. I don't know whether home routers can be configured to do the same with IPQuick service.
Really?