Recently Advanced Rest client is changed to desktop app instead of chrome extension. Earlier browser used to trust the self signed certs but now this is a desktop app so the same work around is not working.
How to trust self signed certs in ARC desktop app?
Update: September 2016
Use the "Use XHR" switch placed in the main toolbar to switch between app's transport implementation and Chrome's. You'll be required to install additional extension that behaves as a proxy between the app and Chrome but you'll be able to pass expired certificates.
Though, I still encourage you to use valid certificate in development environment. With service like letsencrypt.org it can be mostly automated and integrated into development workflow.
Original answer
You can install certificate in Chrome / System and it will work.
There is a tutorial how to do it http://restforchrome.blogspot.co.uk/2016/04/advanced-rest-client.html However there are plenty of tutorials explaining installation on different platforms.
Also I strongly encourage to use services like https://letsencrypt.org/ which provides free certificates. They also provide easy setup scripts for different servers.