Any modest new computer should work well. You don’t need to run Windows, the requirement is Java. You can run most versions of Windows, Linux, or OSX. I don’t think that CE can run the gateway - someone please confirm or correct.
I’m not 100% clear from your post if this computer will be the Ignition gateway and launch clients, or just run clients from another gateway.
For clarification, the mobile module allows you to run clients in an HTML 5 web browser, which targets devices that don’t support the Java Runtime Environment (JRE), such as iPod/iPads and Android phones. What’s happening in the background is the gateway (Windows, Linux, etc) runs and streams the visualization piece to the web browser of mobile clients.