Is it possible to install some kind of software on the mining device with dynamic IP? Or is it an ASIC you don't really have that kind of access to? There are online DNS services like dyndns (and probably others as well, I was a long time since I used these kind of services) where you can install local software that automatically updates your DNS name to point to your dynamic IP. In this case it would be possible to point Awesome Miner to the DNS name instead of the IP.

But in general, just like Axiste point out, Awesome Miner will require you to manually change the IP address.